TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES II: TRIPODS AND COMBINATORIAL CUSPIDALIZATION YUICHIRO HOSHI AND SHINICHI MOCHIZUKI OCTOBER 2021 Abstract. Let Σ be a subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present monograph, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated con- figuration spaces over algebraically closed fields in which the primes of Σ are invertible. The starting point of the theory of the present monograph is a combinatorial anabelian result which, unlike results obtained in previous papers, allows one to eliminate the hypothesis that cuspidal inertia subgroups are preserved by the isomorphism in question. This result allows us to [partially] generalize combinato- rial cuspidalization results obtained in previous papers to the case of outer automorphisms of pro-Σ fundamental groups of configuration spaces that do not necessarily preserve the cuspidal inertia subgroups of the various one-dimensional subquotients of such a fundamental group. Such partial combinatorial cuspidalization results allow one in effect to reduce issues concerning the anabelian geometry of con- figuration spaces to issues concerning the anabelian geometry of hyperbolic curves. These results also allow us, in the case of config- uration spaces of sufficiently large dimension, to give purely group- theoretic characterizations of the cuspidal inertia subgroups of the various one-dimensional subquotients of the pro-Σ fundamental group of a configuration space. We then turn to the study of tripod synchronization, i.e., roughly speaking, the phenomenon that an outer automorphism of the pro-Σ fundamental group of a log config- uration space associated to a stable log curve typically induces the same outer automorphism on the various subquotients of such a fun- damental group determined by tripods [i.e., copies of the projective line minus three points]. Our study of tripod synchronization allows us to show that outer automorphisms of pro-Σ fundamental groups of configuration spaces exhibit somewhat different behavior from the behavior that may be observed — as a consequence of the classical Dehn-Nielsen-Baer theorem — in the case of discrete fundamen- tal groups. Other applications of the theory of tripod synchronization include a result concerning commuting profinite Dehn multi- twists that, a priori, arise from distinct semi-graphs of anabelioids 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H10. Key words and phrases. anabelian geometry, combinatorial anabelian geometry, combinatorial cuspidalization, profinite Dehn twist, tripod, tripod synchronization, Grothendieck-Teichmüller group, semi-graph of anabelioids, hyperbolic curve, con- figuration space. The first author was supported by Grant-in-Aid for Scientific Research (C), No. 24540016, Japan Society for the Promotion of Science. 1 2 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI of pro-Σ PSC-type structures [i.e., the profinite analogue of the notion of a decomposition of a hyperbolic topological surface into hyperbolic subsurfaces, such as “pants”], as well as the computation, in terms of a certain scheme-theoretic fundamental group, of the purely combinatorial/group-theoretic commensurator of the group of profi- nite Dehn multi-twists. Finally, we show that the condition that an outer automorphism of the pro-Σ fundamental group of a stable log curve lift to an outer automorphism of the pro-Σ fundamental group of the corresponding n-th log configuration space, where n ≥ 2 is an integer, is compatible, in a suitable sense, with localization on the dual graph of the stable log curve. This localizability prop- erty, together with the theory of tripod synchronization, is applied to construct a purely combinatorial analogue of the natural outer surjection from the étale fundamental group of the moduli stack of hyperbolic curves over Q to the absolute Galois group of Q. Contents Introduction Notations and Conventions 1. Combinatorial anabelian geometry in the absence of group-theoretic cuspidality 2. Partial combinatorial cuspidalization for F-admissible outomorphisms 3. Synchronization of tripods 4. Glueability of combinatorial cuspidalizations References 2 15 18 32 51 103 167 Introduction Let Σ ⊆ Primes be a subset of the set of prime numbers Primes which is either equal to Primes or of cardinality one. In the present monograph, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated configuration spaces over al- gebraically closed fields in which the primes of Σ are invertible [cf. [MzTa], [CmbCsp], [NodNon], [CbTpI]]. Before proceeding, we review some fundamental notions that play a central role in the present monograph. We shall say that a scheme X over an algebraically closed field k is a semi-stable curve if X is connected and proper over k, and, moreover, for each closed point x of X, the completion of the local ring O X,x is isomorphic over k either to k[[t]] or to k[[t 1 , t 2 ]]/(t 1 t 2 ), where t, t 1 , and t 2 are indeterminates. We shall say that a scheme X over a scheme S is a semi-stable curve if the structure morphism X → S is flat, and, moreover, every geometric fiber of X → S is a semi-stable curve. We shall say that a pair (X, D) consisting of a scheme X over a scheme S and a [possibly empty] closed subscheme D ⊆ X is a pointed stable curve over S if the following COMBINATORIAL ANABELIAN TOPICS II 3 conditions are satisfied: X is a semi-stable curve over S; D is contained in the smooth locus of the structure morphism X → S and étale over S; the invertible sheaf ω X/S (D) — where we write ω X/S for the dualizing sheaf of X/S — is relatively ample [relative to the morphism X → S]. We shall say that a scheme X over a scheme S is a hyperbolic curve over S if there exists a pointed stable curve (Y, E) over S such that Y is smooth over S, and, moreover, X is isomorphic to Y \ E over S. It is well-known [cf. [SGA1], Exposé V, §7] that if X is a connected locally noetherian scheme, and x → X is a geometric point of X, then the category Fét(X) consisting of X-schemes Z whose structure morphism is finite and étale and [necessarily finite étale] X-morphisms forms a Galois category, for which the functor from Fét(X) to the cat- egory of finite sets given by Z → Z × X x is a fundamental functor [cf. [SGA1], Exposé V, Définition 5.1]. Thus, it follows from the general theory of Galois categories [cf. the discussion following [SGA1], Ex- posé V, Remarque 5.10] that one may associate, to the Galois category Fét(X) equipped with the above fundamental functor, the “fundamen- tal pro-group” of the Galois category Fét(X) equipped with the above fundamental functor, which we shall refer to as the étale fundamental group π 1 (X, x) of (X, x). If X is a normal scheme, K is an algebraic closure of the function field K of X, and x is the tautological geometric point of X determined by K, then π 1 (X, x) may be naturally identi- fied with the quotient of Gal(K/K) determined by the union of finite subextensions K ⊆ L ⊆ K such that the normalization of X in L is finite étale over X [cf. [SGA1], Exposé I, Corollaire 10.3]. Since [one verifies easily that] the étale fundamental group is, in a natural sense, independent, up to inner automorphism, of the choice of the basepoint, i.e., the geometric point “x”, we shall omit mention of the basepoint throughout the present monograph. Let G be a topological group. Then we shall write Aut(G) for the group of [continuous] automorphisms of G, Inn(G) ⊆ Aut(G) for the def group of inner automorphisms of G, and Out(G) = Aut(G)/Inn(G) for the group of [continuous] outomorphisms [i.e., outer automorphisms] of G. Thus, an outer automorphism of G is an automorphism of G considered up to composition with an inner automorphism. Let k be a field, k sep a separable closure of k, and X a geometrically def connected scheme of finite type over k. Write G k = Gal(k sep /k) for the absolute Galois group of k. Then it is well-known [cf. [SGA1], Exposé IX, Théorème 6.1] that the natural morphisms of schemes X × k k sep → X → Spec k determine an exact sequence of profinite groups 1 −→ π 1 (X × k k sep ) −→ π 1 (X) −→ G k −→ 1. Write Δ X for the maximal pro-Σ quotient of the étale fundamental group π 1 (X × k k sep ) of X × k k sep and Π X for the quotient of the étale fundamental group π 1 (X) of X by the normal closed subgroup of π 1 (X) 4 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI determined by the kernel of the natural surjection (π 1 (X) ←) π 1 (X × k k sep )  Δ X . Then the above displayed exact sequence determines an exact sequence of profinite groups 1 −→ Δ X −→ Π X −→ G k −→ 1. Next, observe that the above displayed exact sequence induces a natural action of Π X on Δ X by conjugation, i.e., a homomorphism Π X → Aut(Δ X ), which restricts to the tautological homomorphism Δ X → Inn(Δ X ). Thus, by considering the respective quotients by Δ X , we obtain an outer action of G k on Δ X , i.e., a homomorphism G k −→ Out(Δ X ). This outer action is one of the main objects of study in anabelian geometry. In the situation of the preceding paragraph, if X is a hyperbolic curve over k, then each cusp of X [i.e., each geometric point of the smooth compactification of X whose image is not contained in X] determines a conjugacy class of closed subgroups of Δ X [i.e., the inertia subgroup(s) associated to the cusp], each member of which we shall refer to as a cuspidal inertia subgroup of Δ X . Now suppose further that k is the field of fractions of a complete regular local ring R, and that every element of Σ is invertible in R. Suppose, moreover, that X has a stable model over R, i.e., that there exists a pointed stable curve (Y, E) over def S = Spec R such that X is isomorphic to (Y \ E) × R k over k. Then combinatorial anabelian geometry may be described as the study of the combinatorial geometric properties of the irreducible components and nodes [i.e., singular points] of the geometric fiber of (Y, E) over the unique closed point of S by means of the purely group-theoretic properties of the outer action of G k — or, alternatively, various natural subquotients of G k — on Δ X . Here, we observe that this geometric fiber of (Y, E) over the unique closed point of S may be regarded as a sort of degeneration of the hyperbolic curve X. Let k be an algebraically closed field of characteristic ∈ Σ and X a hyperbolic curve over k. For each positive integer m, write • X m for the m-th configuration space of X, i.e., the open sub- scheme of the fiber product of m copies of X over k obtained by removing the various diagonals; • Π m for the maximal pro-Σ quotient of the étale fundamental group π 1 (X m ) of X m ; def def • X 0 = Spec k and Π 0 = {1}. Let n be a positive integer. We shall think of the factors of X n as labeled by the indices 1, . . . , n. Thus, for E ⊆ {1, . . . , n} a subset of cardinality n − m, where m is a nonnegative integer, we have a projection morphism X n → X m obtained by forgetting the factors that belong to E, hence also an induced outer surjection Π n  Π m , i.e., a COMBINATORIAL ANABELIAN TOPICS II 5 surjection considered up to composition with an inner automorphism. Normal closed subgroups Ker(Π n  Π m ) ⊆ Π n obtained in this way will be referred to as fiber subgroups of Π n of length n − m [cf. [MzTa], Definition 2.3, (iii)]. Write X n −→ X n−1 −→ . . . −→ X m −→ . . . −→ X 1 −→ X 0 for the projections obtained by forgetting, successively, the factors la- beled by indices > m [as m ranges over the nonnegative integers ≤ n]. Thus, we obtain a sequence of outer surjections Π n  Π n−1  . . .  Π m  . . .  Π 1  Π 0 . def For each nonnegative integer m ≤ n, write K m = Ker(Π n  Π m ). Thus, we have a filtration of subgroups {1} = K n ⊆ K n−1 ⊆ . . . ⊆ K m ⊆ . . . ⊆ K 1 ⊆ K 0 = Π n . In the situation of the previous paragraph, let Y be a hyperbolic curve over k and Y n a positive integer. Write Y Π Y n for the “Π n ” that occurs in the case where we take “(X, n)” to be (Y, Y n). Let ∼ α : Π n → Y Π Y n be a(n) [continuous] outer isomorphism. Then we shall say that • α is PF-admissible [cf. [CbTpI], Definition 1.4, (i)] if α induces a bijection between the set of fiber subgroups of Π n and the set of fiber subgroups of Y Π Y n ; • α is PC-admissible [cf. [CbTpI], Definition 1.4, (ii), as well as Lemma 3.2, (i), of the present monograph] if, for each positive integer a ≤ n, α(K a ) ⊆ Y Π Y n is a fiber subgroup of Y Π Y n of length Y n − a, and, moreover, the Y Π Y n -conjugacy-orbit of ∼ isomorphisms K a−1 /K a → α(K a−1 )/α(K a ) determined by α induces a bijection between the set of conjugacy classes of cus- pidal inertia subgroups of K a−1 /K a and the set of conjugacy classes of cuspidal inertia subgroups of α(K a−1 )/α(K a ) [where we note that it follows immediately from the various defini- tions involved that the profinite group K a−1 /K a (respectively,. α(K a−1 )/α(K a )) is equipped with a natural structure of pro-Σ surface group — cf. [MzTa], Definition 1.2]; • α is PFC-admissible [cf. [CbTpI], Definition 1.4, (iii)] if α is PF-admissible and PC-admissible. Suppose, moreover, that (X, n) = (Y, Y n), which thus implies that α is a(n) [continuous] outomorphism of Π n = Y Π Y n . Then we shall say that • α is F-admissible [cf. [CmbCsp], Definition 1.1, (ii)] if α(K) = K for every fiber subgroup K of Π n ; • α is C-admissible [cf. [CmbCsp], Definition 1.1, (ii)] if α is PC-admissible, and α(K a ) = K a for each nonnegative integer a ≤ n; 6 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI • α is FC-admissible [cf. [CmbCsp], Definition 1.1, (ii)] if α is F-admissible and C-admissible. One central theme of the present monograph is the issue of n- cuspidalizability [cf. Definition 3.20], i.e., the issue of the extent to which a given isomorphism between the pro-Σ fundamental groups of a pair of hyperbolic curves lifts [necessarily uniquely, up to a per- mutation of factors — cf. [NodNon], Theorem B] to a PFC-admissible [cf. [CbTpI], Definition 1.4, (iii)] isomorphism between the pro-Σ fun- damental groups of the corresponding n-th configuration spaces, for n ≥ 1 a positive integer. In this context, we recall that both the alge- braic and the anabelian geometry of such configuration spaces revolves around the behavior of the various diagonals that are removed from direct products of copies of the given curve in order to construct these configuration spaces. From this point of view, it is perhaps natural to think of the issue of n-cuspidalizability as a sort of abstract profinite analogue of the notion of n-differentiability in the theory of differen- tial manifolds. In particular, it is perhaps natural to think of the theory of the present monograph [as well as of [MzTa], [CmbCsp], [NodNon], [CbTpI]] as a sort of abstract profinite analogue of the classical theory constituted by the differential topology of surfaces. Next, we recall that, to a substantial extent, the theory of combina- torial cuspidalization [i.e., the issue of n-cuspidalizability] developed in [CmbCsp] may be thought of as an essentially formal consequence of the combinatorial anabelian result obtained in [CmbGC], Corol- lary 2.7, (iii). In a similar vein, the generalization of this theory of [CmbCsp] that is summarized in [NodNon], Theorem B, may be re- garded as an essentially formal consequence of the combinatorial an- abelian result given in [NodNon], Theorem A. The development of the theory of the present monograph follows this pattern to a substantial extent. That is to say, in §1, we begin the development of the the- ory of the present monograph by proving a fundamental combinatorial anabelian result [cf. Theorem 1.9], which generalizes the combinato- rial anabelian results given in [CmbGC], Corollary 2.7, (iii); [NodNon], Theorem A. A substantial portion of the main results obtained in the remainder of the present monograph may be understood as consisting of various applications of Theorem 1.9. By comparison to the combinatorial anabelian results of [CmbGC], Corollary 2.7, (iii); [NodNon], Theorem A, the main technical feature of the combinatorial anabelian result given in Theorem 1.9 of the present monograph is that it allows one, to a substantial extent, to eliminate the group-theoretic cuspidality hypothesis — i.e., the assumption to the effect that the isomorphism between pro- Σ fundamental groups of stable log curves under consideration [that is to say, in effect, an isomorphism between the pro-Σ fundamental groups COMBINATORIAL ANABELIAN TOPICS II 7 of certain degenerations of hyperbolic curves] necessarily preserves cus- pidal inertia subgroups — that plays a central role in the proofs of ear- lier combinatorial anabelian results. In §2, we apply Theorem 1.9 to obtain the following [partial] combinatorial cuspidalization result [cf. Theorem 2.3, (i), (ii); Corollary 3.22], which [partially] generalizes [NodNon], Theorem B. Theorem A (Partial combinatorial cuspidalization for F-ad- missible outomorphisms). Let (g, r) be a pair of nonnegative inte- gers such that 2g − 2 + r > 0; n a positive integer; Σ a set of prime numbers which is either equal to the set of all prime numbers or of car- dinality one; X a hyperbolic curve of type (g, r) over an algebraically closed field of characteristic ∈ Σ; X n the n-th configuration space of X; Π n the maximal pro-Σ quotient of the fundamental group of X n ; Out F (Π n ) ⊆ Out(Π n ) the subgroup of F-admissible outomorphisms [i.e., roughly speaking, outer automorphisms that preserve the fiber subgroups — cf. the dis- cussion preceding Theorem A; [CmbCsp], Definition 1.1, (ii), for more details] of Π n ; Out FC (Π n ) ⊆ Out F (Π n ) the subgroup of FC-admissible outomorphisms [i.e., roughly speak- ing, outer automorphisms that preserve the fiber subgroups and the cuspidal inertia subgroups — cf. the discussion preceding Theorem A; [CmbCsp], Definition 1.1, (ii), for more details] of Π n . Then the fol- lowing hold: (i) Write  def n inj = 1 2 if r  = 0, if r = 0 ,  def n bij = 3 4 if r  = 0, if r = 0 . If n ≥ n inj (respectively, n ≥ n bij ), then the natural homomor- phism Out F (Π n+1 ) −→ Out F (Π n ) induced by the projections X n+1 → X n obtained by forgetting any one of the n + 1 factors of X n+1 [cf. [CbTpI], Theorem A, (i)] is injective (respectively, bijective). (ii) Write ⎧ ⎨ 2 def 3 n FC = ⎩ 4 if (g, r) = (0, 3), if (g, r)  = (0, 3) and r  = 0, if r = 0 . If n ≥ n FC , then it holds that Out FC (Π n ) = Out F (Π n ) . 8 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (iii) Suppose that (g, r) ∈ {(0, 3); (1, 1)}. Then the natural injec- tion [cf. [NodNon], Theorem B] Out FC (Π 2 ) → Out FC (Π 1 ) induced by the projections X 2 → X 1 obtained by forgetting ei- ther of the two factors of X 2 is not surjective. Here, we remark that the non-surjectivity discussed in Theorem A, (iii), is, in fact, obtained as a consequence of the theory of tripod syn- chronization developed in §3 [cf. the discussion preceding Theorem C below]. This non-surjectivity is remarkable in that it yields an impor- tant example of substantially different behavior in the theory of profi- nite fundamental groups of hyperbolic curves from the corresponding theory in the discrete case. That is to say, in the case of the classical discrete fundamental group of a hyperbolic topological surface, the sur- jectivity of the corresponding homomorphism may be derived as an essentially formal consequence of the well-known Dehn-Nielsen-Baer theorem in the theory of topological surfaces [cf. the discussion of Re- mark 3.22.1, (i)]. In particular, it constitutes an important “counterex- ample” to the “line of reasoning” [i.e., for instance, of the sort which appears in the final paragraph of [Lch], §1; the discussion between [Lch], Theorem 5.1, and [Lch], Conjecture 5.2] that one should expect essentially analogous behavior in the theory of profinite fundamental groups of hyperbolic curves to the relatively well understood behav- ior observed classically in the theory of discrete fundamental groups of topological surfaces [cf. the discussion of Remark 3.22.1, (iii)]. Theorem A leads naturally to the following strengthening of the result obtained in [CbTpI], Theorem A, (ii), concerning the group- theoreticity of the cuspidal inertia subgroups of the various one- dimensional subquotients of a configuration space group [cf. Corol- lary 2.4]. Theorem B (PFC-admissibility of outomorphisms). In the no- tation of Theorem A, write Out PF (Π n ) ⊆ Out(Π n ) for the subgroup of PF-admissible outomorphisms [i.e., roughly speak- ing, outer automorphisms that preserve the fiber subgroups up to a pos- sible permutation of the factors — cf. the discussion preceding Theo- rem A; [CbTpI], Definition 1.4, (i), for more details] and Out PFC (Π n ) ⊆ Out PF (Π n ) for the subgroup of PFC-admissible outomorphisms [i.e., roughly speak- ing, outer automorphisms that preserve the fiber subgroups and the cus- pidal inertia subgroups up to a possible permutation of the factors — cf. the discussion preceding Theorem A; [CbTpI], Definition 1.4, (iii), COMBINATORIAL ANABELIAN TOPICS II 9 for more details]. Let us regard the symmetric group on n letters S n as a subgroup of Out(Π n ) via the natural inclusion S n → Out(Π n ) obtained by permuting the various factors of X n . Finally, suppose that (g, r) ∈ {(0, 3); (1, 1)}. Then the following hold: (i) We have an equality Out(Π n ) = Out PF (Π n ). If, moreover, (r, n)  = (0, 2), then we have equalities Out(Π n ) = Out PF (Π n ) = Out F (Π n ) × S n . (ii) If either r > 0, n ≥ 3 or n ≥ 4, then we have equalities Out(Π n ) = Out PFC (Π n ) = Out FC (Π n ) × S n . The partial combinatorial cuspidalization of Theorem A has natural applications to the relative and [semi-]absolute anabelian geom- etry of configuration spaces [cf. Corollaries 2.5, 2.6], which gen- eralize the theory of [AbsTpI], §1. Roughly speaking, these results allow one, in a wide variety of cases, to reduce issues concerning the relative and [semi-]absolute anabelian geometry of configuration spaces to the corresponding issues concerning the relative and [semi-]absolute anabelian geometry of hyperbolic curves. Also, we remark that in this context, we obtain a purely scheme-theoretic result [cf. Lemma 2.7] that states, roughly speaking, that the theory of isomorphisms [of schemes!] between configuration spaces associated to hyperbolic curves may be reduced to the theory of isomorphisms [of schemes!] between hyper- bolic curves. In §3, we take up the study of [the group-theoretic versions of] the various tripods [i.e., copies of the projective line minus three points] that occur in the various one-dimensional fibers of the log configura- tion spaces associated to a stable log curve [cf. the discussion entitled “Curves” in [CbTpI], §0]. Roughly speaking, these tripods either oc- cur in the original stable log curve or arise as the result of blowing up various cusps or nodes that occur in the one-dimensional fibers of log configuration spaces of lower dimension [cf. Figure 1 at the end of the present Introduction]. In fact, a substantial portion of §3 is devoted precisely to the theory of classification of the various tripods that occur in the one-dimensional fibers of the log configuration spaces associated to a stable log curve [cf. Lemmas 3.6, 3.8]. This leads natu- rally to the study of the phenomenon of tripod synchronization, i.e., roughly speaking, the phenomenon that an outomorphism [that is to 10 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI say, an outer automorphism] of the pro-Σ fundamental group of a log configuration space associated to a stable log curve typically induces the same outer automorphism on the various [group-theoretic] tripods that occur in subquotients of such a fundamental group [cf. Theo- rems 3.16, 3.17, 3.18]. The phenomenon of tripod synchronization, in turn, leads naturally to the definition of the tripod homomorphism [cf. Definition 3.19], which may be thought of as the homomorphism obtained by associating to an [FC-admissible] outer automorphism of the pro-Σ fundamental group of the n-th log configuration space as- sociated to a stable log curve, where n ≥ 3 is a positive integer, the outer automorphism induced on a [group-theoretic] central tripod, i.e., roughly speaking, a tripod that arises, in the case where n = 3 and the given stable log curve has no nodes, by blowing up the intersection of the three diagonal divisors of the direct product of three copies of the curve. Theorem C (Synchronization of tripods in three or more di- mensions). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; n a positive integer; Σ a set of prime numbers which is either equal to the set of all prime numbers or of cardinality one; k an algebraically closed field of characteristic ∈ Σ; (Spec k) log the log scheme obtained by equipping Spec k with the log structure determined by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over (Spec k) log . Write G for the semi-graph of anabelioids of pro-Σ PSC-type determined by the stable log curve X log . For each positive integer i, write X i log for the i-th log configuration space of the stable log curve X log [cf. the discussion entitled “Curves” in “Notations and Conventions”]; Π i for the maximal pro-Σ quotient of the kernel of the natural surjection π 1 (X i log )  π 1 ((Spec k) log ). Let T ⊆ Π m be a {1, · · · , m}-tripod of Π n [cf. Definition 3.3, (i)] for m a positive integer ≤ n. Suppose that n ≥ 3. Let Π tpd ⊆ Π 3 be a 1-central {1, 2, 3}-tripod of Π n [cf. Definitions 3.3, (i); 3.7, (ii)]. Then the following hold: (i) The commensurator and centralizer of T in Π m satisfy the equality C Π m (T ) = T × Z Π m (T ) . Thus, if an outomorphism α of Π m preserves the Π m -conjugacy class of T ⊆ Π m , then one obtains a “restriction” α| T ∈ Out(T ). (ii) Let α ∈ Out FC (Π n ) be an FC-admissible outomorphism of Π n . Then the outomorphism of Π 3 induced by α preserves the Π 3 - conjugacy class of Π tpd ⊆ Π 3 . In particular, by (i), we obtain COMBINATORIAL ANABELIAN TOPICS II 11 a natural homomorphism T Π tpd : Out FC (Π n ) −→ Out(Π tpd ) . We shall refer to this homomorphism as the tripod homo- morphism associated to Π n . (iii) Let α ∈ Out FC (Π n ) be an FC-admissible outomorphism of Π n such that the outomorphism α m of Π m induced by α preserves the Π m -conjugacy class of T ⊆ Π m and induces [cf. (i)] the identity automorphism of the set of T -conjugacy classes of cuspidal inertia subgroups of T . Then there exists a geometric ∼ [cf. Definition 3.4, (ii)] outer isomorphism Π tpd → T with respect to which the outomorphism T Π tpd (α) ∈ Out(Π tpd ) [cf. (ii)] is compatible with the outomorphism α m | T ∈ Out(T ) [cf. (i)]. (iv) Suppose, moreover, that either n ≥ 4 or r  = 0. Then the homomorphism T Π tpd of (ii) factors through Out C (Π tpd ) Δ+ ⊆ Out(Π tpd ) [cf. Definition 3.4, (i)], and, moreover, the resulting homomorphism T Π tpd : Out F (Π n ) = Out FC (Π n ) −→ Out C (Π tpd ) Δ+ [cf. Theorem A, (ii)] is surjective. Here, we remark that the surjectivity of the tripod homomorphism [cf. Theorem C, (iv)] is obtained [cf. Corollary 4.15] as a consequence of the theory of glueability of combinatorial cuspidalizations developed in §4 [cf. the discussion preceding Theorem F below]. Also, we recall that the codomain of this surjective tripod homomorphism Out C (Π tpd ) Δ+ may be identified with the [pro-Σ] Grothendieck-Teichmüller group GT Σ [cf. the discussion of [CmbCsp], Remark 1.11.1]. Since GT Σ may be thought of as a sort of abstract combinatorial approximation of the absolute Galois group G Q of the rational number field Q, it is thus natural to think of the surjective tripod homomorphism Out F (Π n )  Out C (Π tpd ) Δ+ of Theorem C, (iv), as a sort of abstract combinatorial version of the natural surjective outer homomorphism π 1 ((M g,[r] ) Q )  G Q induced on étale fundamental groups by the structure morphism (M g,[r] ) Q → Spec (Q) of the moduli stack (M g,[r] ) Q of hyperbolic curves of type (g, r) [cf. the discussion of Remark 3.19.1]. In particular, the kernel of the tripod homomorphism — which we denote by Out F (Π n ) geo 12 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI — may be thought of as a sort of abstract combinatorial analogue of the geometric étale fundamental group of (M g,[r] ) Q [i.e., the kernel of the natural outer homomorphism π 1 ((M g,[r] ) Q )  G Q ]. One interesting application of the theory of tripod synchronization is the following. Fix a pro-Σ fundamental group of a hyperbolic curve. Recall the notion of a nondegenerate profinite Dehn multi-twist [cf. [CbTpI], Definition 4.4; [CbTpI], Definition 5.8, (ii)] associated to a structure of semi-graph of anabelioids of pro-Σ PSC-type on such a fun- damental group. Here, we recall that such a structure may be thought of as a sort of profinite analogue of the notion of a decomposition of a hyperbolic topological surface into hyperbolic subsurfaces [i.e., such as “pants”]. Then the following result asserts that, under certain techni- cal conditions, any such nondegenerate profinite Dehn multi-twist that commutes with another nondegenerate profinite Dehn multi-twist as- sociated to some given totally degenerate semi-graph of anabelioids of pro-Σ PSC-type [cf. [CbTpI], Definition 2.3, (iv)] necessarily arises from a structure of semi-graph of anabelioids of pro-Σ PSC-type that is “co-Dehn” to, i.e., arises by applying a deformation to, the given totally degenerate semi-graph of anabelioids of pro-Σ PSC-type [cf. Corollary 3.25]. This sort of result is reminiscent of topological results concerning subgroups of the mapping class group generated by pairs of positive Dehn multi-twists [cf. [Ishi], [HT]]. Theorem D (Co-Dehn-ness of degeneration structures in the totally degenerate case). In the notation of Theorem C, for i = 1, 2, let Y i log be a stable log curve over (Spec k) log ; H i the “G” that occurs in the case where we take “X log ” to be Y i log ; (H i , S i , φ i ) a 3- cuspidalizable degeneration structure on G [cf. Definition 3.23, (i), (v)]; α i ∈ Out(Π G ) a nondegenerate (H i , S i , φ i )-Dehn multi-twist of G [cf. Definition 3.23, (iv)]. Suppose that α 1 commutes with α 2 , and that H 2 is totally degenerate [cf. [CbTpI], Definition 2.3, (iv)]. Suppose, moreover, that one of the following conditions is satisfied: (i) r  = 0. (ii) α 1 and α 2 are positive definite [cf. Definition 3.23, (iv)]. Then (H 1 , S 1 , φ 1 ) is co-Dehn to (H 2 , S 2 , φ 2 ) [cf. Definition 3.23, (iii)], or, equivalently [since H 2 is totally degenerate], (H 2 , S 2 , φ 2 ) (H 1 , S 1 , φ 1 ) [cf. Definition 3.23, (ii)]. Another interesting application of the theory of tripod synchroniza- tion is to the computation, in terms of a certain scheme-theoretic fundamental group, of the purely combinatorial commensurator of the subgroup of profinite Dehn multi-twists in the group of 3-cuspidali- zable, FC-admissible, “geometric” outer automorphisms of the pro-Σ COMBINATORIAL ANABELIAN TOPICS II 13 fundamental group of a totally degenerate stable log curve [cf. Corol- lary 3.27]. Here, we remark that the scheme-theoretic [or, perhaps more precisely, “log algebraic stack-theoretic”] fundamental group that ap- pears is, roughly speaking, the pro-Σ geometric fundamental group of a formal neighborhood, in the corresponding logarithmic moduli stack, of the point determined by the given totally degenerate sta- ble log curve. In particular, this computation may also be regarded as a sort of purely combinatorial algorithm for constructing this scheme-theoretic fundamental group [cf. Remark 3.27.1]. Theorem E (Commensurator of profinite Dehn multi-twists in the totally degenerate case). In the notation of Theorem C [so n ≥ 3], suppose further that if r = 0, then n ≥ 4. Also, we assume that G is totally degenerate [cf. [CbTpI], Definition 2.3, def (iv)]. Write s : Spec k → (M g,[r] ) k = (M g,[r] ) Spec k [cf. the discussion entitled “Curves” in “Notations and Conventions”] for the underly- ing (1-)morphism of algebraic stacks of the classifying (1-)morphism log log def (Spec k) log → (M g,[r] ) k = (M g,[r] ) Spec k [cf. the discussion entitled “Curves” in “Notations and Conventions”] of the stable log curve X log  log for the log scheme obtained by equipping N  s def over (Spec k) log ; N = s Spec k with the log structure induced, via s, by the log structure of log (M g,[r] ) k ; N s log for the log stack obtained by forming the [stack-theoretic]  log by the natural action of the finite k- quotient of the log scheme N s group “s × (M g,[r] ) k s”, i.e., the fiber product over (M g,[r] ) k of two copies of s; N s for the underlying stack of the log stack N s log ; I N s ⊆ π 1 (N s log ) for the closed subgroup of the log fundamental group π 1 (N s log ) of N s log given by the kernel of the natural surjection π 1 (N s log )  π 1 (N s ) [in- duced by the (1-)morphism N s log → N s obtained by forgetting the log (Σ) structure]; π 1 (N s log ) for the quotient of π 1 (N s log ) by the kernel of the Σ natural surjection from I N s to its maximal pro-Σ quotient I N . Then s we have an equality N Out F (Π n ) geo (Dehn(G)) = C Out F (Π n ) geo (Dehn(G)) and a natural commutative diagram of profinite groups 1 −−−→ Σ I N ⏐ s ⏐  −−−→ (Σ) π 1 (N s log ) ⏐ ⏐  −−−→ π 1 (N s ) −−−→ 1 ⏐ ⏐  1 −−−→ Dehn(G) −−−→ C Out F (Π n ) geo (Dehn(G)) −−−→ Aut(G) −−−→ 1 [cf. Definition 3.1, (ii), concerning the notation “G”] — where the horizontal sequences are exact, and the vertical arrows are isomor- phisms. Moreover, Dehn(G) is open in C Out F (Π n ) geo (Dehn(G)). 14 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI In §4, we show, under suitable technical conditions, that an auto- morphism of the pro-Σ fundamental group of the log configuration space associated to a stable log curve necessarily preserves the graph- theoretic structure of the various one-dimensional fibers of such a log configuration space [cf. Theorem 4.7]. This allows us to verify the glueability of combinatorial cuspidalizations, i.e., roughly speak- ing, that, for n ≥ 2 a positive integer, the datum of an n-cuspidalizable outer automorphism of the pro-Σ fundamental group of a stable log curve is equivalent, up to possible composition with a profinite Dehn multi-twist, to the datum of a collection of n-cuspidalizable automor- phisms of the pro-Σ fundamental groups of the various irreducible com- ponents of the given stable log curve that satisfy a certain gluing condi- tion involving the induced outer actions on tripods [cf. Theorem 4.14]. Theorem F (Glueability of combinatorial cuspidalizations). In the notation of Theorem C, write Out FC (Π n ) brch ⊆ Out FC (Π n ) for the closed subgroup of Out FC (Π n ) consisting of FC-admissible out- omorphisms α of Π n such that the outomorphism of Π 1 determined by α induces the identity automorphism of Vert(G), Node(G), and, more- over, fixes each of the branches of every node of G [cf. Definition 4.6, (i)]; Out FC ((Π v ) n ) Glu(Π n ) ⊆ v∈Vert(G) for the closed subgroup of v∈Vert(G) Out FC ((Π v ) n ) consisting of “glue- able” collections of outomorphisms of the groups “(Π v ) n ” [cf. Defini- tion 4.9, (iii)]. Then we have a natural exact sequence of profinite groups 1 −→ Dehn(G) −→ Out FC (Π n ) brch −→ Glu(Π n ) −→ 1 . This glueability result may, alternatively, be thought of as a re- sult that asserts the localizability [i.e., relative to localization on the dual semi-graph of the given stable log curve] of the notion of n- cuspidalizability. In this context, it is of interest to observe that this glueability result may be regarded as a natural generalization, to the case of n-cuspidalizability for n ≥ 2, of the glueability result obtained in [CbTpI], Theorem B, (iii), in the “1-cuspidalizable” case, which is derived as a consequence of the theory of localizability [i.e., relative to localization on the dual semi-graph of the given stable log curve] and synchronization of cyclotomes developed in [CbTpI], §3, §4. From this point of view, it is also of interest to observe that the sufficiency portion of [the equivalence that constitutes] this glueability result [i.e., Theorem F] may be thought of as a sort of “converse” to the theory of tripod synchronizations developed in §3 [i.e., of which the necessity COMBINATORIAL ANABELIAN TOPICS II 15 portion of this glueability result is, in essence, a formal consequence — cf. the proof of Lemma 4.10, (ii)]. Indeed, the bulk of the proof given in §4 of Theorem 4.14 is devoted to the sufficiency portion of this result, which is verified by means of a detailed combinatorial analysis [cf. the proof of [CbTpI], Proposition 4.10, (ii)] of the noncyclically primi- tive and cyclically primitive cases [cf. Lemmas 4.12, 4.13; Figures 2, 3, 4]. Finally, we apply this glueability result to derive a cuspidalization theorem — i.e., in the spirit of and generalizing the corresponding results of [AbsCsp], Theorem 3.1; [Hsh], Theorem 0.1; [Wkb], Theorem C [cf. Remark 4.16.1] — for geometrically pro-l fundamental groups of stable log curves over finite fields [cf. Corollary 4.16]. That is to say, in the case of stable log curves over finite fields, the condition of compatibility with the Galois action is sufficient to imply the n-cuspidalizability of arbi- trary isomorphisms between the geometric pro-l fun- damental groups, for n ≥ 1. In this context, it is of interest to recall that strong anabelian results [i.e., in the style of the “Grothendieck Conjecture”] for such geomet- rically pro-l fundamental groups of stable log curves over finite fields are not known in general, at the time of writing. On the other hand, we observe that in the case of totally degenerate stable log curves over finite fields, such “strong anabelian results” may be obtained un- der certain technical conditions [cf. Corollary 4.17; Remarks 4.17.1, 4.17.2]. Notations and Conventions Sets: If S is a set, then we shall denote by #S the cardinality of S. Groups: We shall refer to an element of a group as trivial (respectively, nontrivial) if it is (respectively, is not) equal to the identity element of the group. We shall refer to a nonempty subset of a group as trivial (respectively, nontrivial) if it is (respectively, is not) equal to the set whose unique element is the identity element of the group. Topological groups: Let G be a topological group and J, H ⊆ G closed subgroups. Then we shall write def Z J (H) = { j ∈ J | jh = hj for any h ∈ H } = Z G (H) ∩ J for the centralizer of H in J, def Z(G) = Z G (G) for the center of G, and def Z J loc (H) = lim Z J (U ) ⊆ J −→ 16 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI — where the inductive limit is over all open subgroups U ⊆ H of H — def for the “local centralizer” of H in J. We shall write Z loc (G) = Z G loc (G) for the “local center” of G. Thus, a profinite group G is slim [cf. the discussion entitled “Topological groups” in [CbTpI], §0] if and only if Z loc (G) = {1}. Rings: If R is a commutative ring with unity, then we shall write R ∗ for the multiplicative group of invertible elements of R. Curves: Let g, r 1 , r 2 be nonnegative integers such that 2g − 2 + r 1 + r 2 > 0. Then we shall write M g,[r 1 ]+r 2 for the moduli stack of pointed stable curves of type (g, r 1 + r 2 ), where the first r 1 marked points are regarded as unordered, but the last r 2 marked points are regarded as ordered, over Z; M g,[r 1 ]+r 2 ⊆ M g,[r 1 ]+r 2 for the open sub- log stack of M g,[r 1 ]+r 2 that parametrizes smooth curves; M g,[r 1 ]+r 2 for the log stack obtained by equipping M g,[r 1 ]+r 2 with the log structure as- sociated to the divisor with normal crossings M g,[r 1 ]+r 2 \ M g,[r 1 ]+r 2 ⊆ M g,[r 1 ]+r 2 ; C g,[r 1 ]+r 2 → M g,[r 1 ]+r 2 for the tautological stable curve over M g,[r 1 ]+r 2 ; D g,[r 1 ]+r 2 ⊆ C g,[r 1 ]+r 2 for the corresponding tautological di- visor of cusps of C g,[r 1 ]+r 2 → M g,[r 1 ]+r 2 . Then the divisor given by the union of D g,[r 1 ]+r 2 with the inverse image in C g,[r 1 ]+r 2 of the divi- sor M g,[r 1 ]+r 2 \ M g,[r 1 ]+r 2 ⊆ M g,[r 1 ]+r 2 determines a log structure on log C g,[r 1 ]+r 2 ; write C g,[r 1 ]+r 2 for the resulting log stack. Thus, we obtain log log a (1-)morphism of log stacks C g,[r 1 ]+r 2 → M g,[r 1 ]+r 2 . We shall write log C g,[r 1 ]+r 2 ⊆ C g,[r 1 ]+r 2 for the interior of C g,[r 1 ]+r 2 [cf. the discussion entitled “Log schemes” in [CbTpI], §0]. In particular, we obtain a (1-)morphism of stacks C g,[r 1 ]+r 2 → M g,[r 1 ]+r 2 . Moreover, for a nonneg- def ative integer r such that 2g−2+r > 0, we shall write M g,[r] = M g,[r]+0 ; log def def log def def M g,[r] = M g,[r]+0 ; M g,[r] = M g,[r]+0 ; C g,[r] = C g,[r]+0 ; D g,[r] = D g,[r]+0 ; log def log def C g,[r] = C g,[r]+0 ; C g,[r] = C g,[r]+0 . In particular, the stack M g,[r] may be regarded as a moduli stack of hyperbolic curves of type (g, r) over Z. If S is a scheme, then we shall denote by means of a subscript S the result of base-changing via the structure morphism S → Spec Z the various log stacks of the above discussion. Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; n a positive integer; X log a stable log curve [cf. the discussion entitled “Curves” in [CbTpI], §0] of type (g, r) over a log scheme S log . Then we shall refer to the log scheme obtained by pulling back the (1-)morphism log log M g,[r]+n → M g,[r] given by forgetting the last n [ordered] points via the classifying (1-)morphism S log → M g,[r] of X log as the n-th log conguration space of X log . COMBINATORIAL ANABELIAN TOPICS II 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . tripod . . . . . . . . . tripod tripod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tripod tripod tripod Figure 1 : tripods in the various fibers of a configuration space 18 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 1. Combinatorial anabelian geometry in the absence of group-theoretic cuspidality In the present §1, we discuss various combinatorial versions of the Grothendieck Conjecture for outer representations of NN- and IPSC- type [cf. Theorem 1.9 below]. These Grothendieck Conjecture-type results may be regarded as generalizations of [NodNon], Corollary 4.2; [NodNon], Remark 4.2.1, that may be applied to isomorphisms that are not necessarily group-theoretically cuspidal. For instance, we prove [cf. Theorem 1.9, (ii), below] that any isomorphism between outer representations of IPSC-type [cf. [NodNon], Definition 2.4, (i)] is nec- essarily group-theoretically verticial, i.e., roughly speaking, preserves the verticial subgroups. A basic reference for the theory of semi-graphs of anabelioids of PSC- type is [CmbGC]. We shall use the terms “semi-graph of anabelioids of PSC-type”, “PSC-fundamental group of a semi-graph of anabelioids of PSC-type”, “finite étale covering of semi-graphs of anabelioids of PSC- type”, “vertex”, “edge”, “node”, “cusp”, “verticial subgroup”, “edge-like subgroup”, “nodal subgroup”, “cuspidal subgroup”, and “sturdy” as they are defined in [CmbGC], Definition 1.1 [cf. also Remark 1.1.2 below]. Also, we shall apply the various notational conventions established in [NodNon], Definition 1.1, and refer to the “PSC-fundamental group of a semi-graph of anabelioids of PSC-type” simply as the “fundamental group” [of the semi-graph of anabelioids of PSC-type]. That is to say, we shall refer to the maximal pro-Σ quotient of the fundamental group of a semi-graph of anabelioids of pro-Σ PSC-type [as a semi- graph of anabelioids!] as the “fundamental group of the semi-graph of anabelioids of PSC-type”. In the present §1, let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Write G for the under- lying semi-graph of G, Π G for the [pro-Σ] fundamental group of G, and G  → G for the universal covering of G corresponding to Π G . Then since the fundamental group Π G of G is topologically finitely generated, the profinite topology of Π G induces [profinite] topologies on Aut(Π G ) and Out(Π G ) [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. If, moreover, we write Aut(G) for the automorphism group of G, then, by the discussion preceding [CmbGC], Lemma 2.1, the natural homomorphism Aut(G) −→ Out(Π G ) is an injection with closed image. [Here, we recall that an automor- phism of a semi-graph of anabelioids consists of an automorphism of the underlying semi-graph, together with a compatible system of iso- morphisms between the various anabelioids at each of the vertices and COMBINATORIAL ANABELIAN TOPICS II 19 edges of the underlying semi-graph which are compatible with the var- ious morphisms of anabelioids associated to the branches of the under- lying semi-graph — cf. [SemiAn], Definition 2.1; [SemiAn], Remark 2.4.2.] Thus, by equipping Aut(G) with the topology induced via this homomorphism by the topology of Out(Π G ), we may regard Aut(G) as being equipped with the structure of a profinite group. Definition 1.1. We shall say that an element γ ∈ Π G of Π G is verticial (respectively, edge-like; nodal; cuspidal) if γ is contained in a verticial (respectively, an edge-like; a nodal; a cuspidal) subgroup of Π G . Remark 1.1.1. Let γ ∈ Π G be a nontrivial [cf. the discussion entitled “Groups” in “Notations and Conventions”] element of Π G . If γ ∈ Π G is edge-like [cf. Definition 1.1], then it follows from [NodNon], Lemma 1.5,  such that γ ∈ Π e  . If γ ∈ Π G that there exists a unique edge e  ∈ Edge( G) is verticial, but not nodal [cf. Definition 1.1], then it follows from  [NodNon], Lemma 1.9, (i), that there exists a unique vertex v  ∈ Vert( G) such that γ ∈ Π v  . Remark 1.1.2. Here, we take the opportunity to correct an unfortu- nate misprint in [CmbGC]. In the final sentence of [CmbGC], Definition 1.1, (ii), the phrase “rank ≥ 2” should read “rank > 2”. In particular, we shall say that G is sturdy if the abelianization of the image, in the quotient Π G  Π unr of Π G by the normal closed subgroup normally G topologically generated by the edge-like subgroups, of every verticial subgroup of Π G is free of rank > 2 over Z Σ . Here, we note in passing that G is sturdy if and only if every vertex of G is of genus ≥ 2 [cf. [CbTpI], Definition 2.3, (iii)]. Lemma 1.2 (Existence of a certain connected finite étale cov- ering). Let n be a positive integer which is a product [possibly with  v  ∈ Vert( G).  Write multiplicities!] of primes ∈ Σ; e  1 , e  2 ∈ Edge( G); def def def e 1 = e  1 (G), e 2 = e  2 (G), and v = v  (G). Suppose that the following conditions are satisfied: (i) G is untangled [cf. [NodNon], Definition 1.2]. (ii) If e 1 is a node, then the following condition holds: Let w, w  ∈ V(e 1 ) be the two distinct elements of V(e 1 ) [cf. (i)]. Then #(N (w) ∩ N (w  )) ≥ 3. (iii) If e 1 is a cusp, then the following condition holds: Let w ∈ V(e 1 ) be the unique element of V(e 1 ). Then #C(w) ≥ 3. 20 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (iv) e 1  = e 2 . (v) v ∈ V(e 1 ). Then there exists a finite étale Galois subcovering G  → G of G  → G such that n divides [Π e  1 : Π e  1 ∩ Π G  ], and, moreover, Π e  2 , Π v  ⊆ Π G  . Proof. Suppose that e 1 is a node (respectively, cusp). Write H for the [uniquely determined] sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of G whose set of vertices is = V(e 1 ) = {w, w  } [cf. condition (ii)] (respectively, = {w} [cf. condition (iii)]). Now it follows from condition (ii) (respectively, (iii)) that there exists an e 3 ∈ Node(G| H ) = N (w) ∩ N (w  ) (respectively, ∈ Cusp(G| H ) ∩ Cusp(G) = C(w)) [cf. [CbTpI], Definition 2.2, (ii)] such that e 3  = e 2 . Moreover, again by applying condition (ii) (respectively, (iii)), together with the well-known structure of the abelianization of the fundamental group of a smooth curve over an algebraically closed field of characteristic ∈ Σ, we conclude that there exists a finite étale Galois covering G H  → G| H that arises from a normal open subgroup of Π G| H and which is unramified at every element of Edge(G| H ) \ {e 1 , e 3 } and totally ramified at e 1 , e 3 with ramification indices divisible by n. Now since G H  → G| H is unramified at every element of Cusp(G| H ) ∩ Node(G), one may extend this covering to a finite étale Galois subcovering G  → G of G  → G which restricts to the trivial covering over every vertex u of G such that u  = w, w  (respectively, u  = w). Moreover, it follows immediately from the construction of G  → G that n divides [Π e  1 : Π e  1 ∩ Π G  ], and Π e  2 , Π v  ⊆ Π G  . This completes the proof of Lemma 1.2.  Lemma 1.3 (Product of edge-like elements). Let γ 1 , γ 2 ∈ Π G be two nontrivial edge-like elements of Π G [cf. Definition 1.1]. Write  for the unique elements of Edge( G)  such that γ 1 ∈ e  1 , e  2 ∈ Edge( G) Π e  1 , γ 2 ∈ Π e  2 [cf. Remark 1.1.1]. Suppose that the following conditions are satisfied: (i) For every positive integer n, it holds that γ 1 n γ 2 n is verticial. (ii) e  1  = e  2 . Then there exists a [necessarily unique — cf. [NodNon], Remark 1.8.1,  such that { v ); in particular, it holds that (iii)] v  ∈ Vert( G) e 1 , e  2 } ⊆ E( γ 1 γ 2 ∈ Π v  . Proof. Since e  1  = e  2 [cf. condition (ii)], one verifies easily that there exists a finite étale Galois subcovering H → G of G  → G that satisfies the following conditions: (1) e  1 (H)  = e  2 (H). COMBINATORIAL ANABELIAN TOPICS II 21 (2) H is untangled [cf. [NodNon], Definition 1.2; [NodNon], Re- mark 1.2.1, (i), (ii)].  then the following holds: Let (3) For i ∈ {1, 2}, if e  i ∈ Node( G), e i (H)) be the two distinct elements of V( e i (H)) [cf. w, w  ∈ V( (ii)]. Then #(N (w) ∩ N (w  )) ≥ 3.  then the following holds: (4) For i ∈ {1, 2}, if e  i ∈ Cusp( G), Let w ∈ V( e i (H)) be the unique element of V( e i (H)). Then #C(w) ≥ 3. Now it is immediate that there exists a positive integer m such that  be such that γ 1 m γ 2 m ∈ γ 1 m ∈ Π e  1 ∩ Π H , γ 2 m ∈ Π e  2 ∩ Π H . Let v  ∈ Vert( G) Π v  [cf. condition (i)]. Suppose that v  (H) ∈ V( e 1 (H)). Then it follows from Lemma 1.2 that there exists a finite étale Galois subcovering H  → H of G  → H such that γ 1 m ∈ Π H  , and, moreover, Π e  2 ∩Π H , Π v  ∩Π H ⊆ Π H  . But this implies that γ 2 m , γ 1 m γ 2 m ∈ Π H  , hence that γ 1 m ∈ Π H  , a contradiction. In particular, it holds that v  (H) ∈ V( e 1 (H)); a similar argument implies e 1 (H)) ∩ V( e 2 (H))  = ∅. Thus, by that v  (H) ∈ V( e 2 (H)), hence that V( applying this argument to a suitable system of connected finite étale coverings of H, we conclude that V( e 1 )∩V( e 2 )  = ∅, i.e., that there exists  such that { a v  ∈ Vert( G) e 1 , e  2 } ⊆ E( v ). Then since Π e  1 , Π e  2 ⊆ Π v  , it follows immediately that γ 1 γ 2 ∈ Π v  . This completes the proof of Lemma 1.3.  Proposition 1.4 (Group-theoretic characterization of closed subgroups of edge-like subgroups). Let H ⊆ Π G be a closed sub- group of Π G . Then the following conditions are equivalent: (i) H is contained in an edge-like subgroup. (ii) An open subgroup of H is contained in an edge-like sub- group. (iii) Every element of H is edge-like [cf. Definition 1.1]. (iv) There exists a connected finite étale subcovering G † → G of G  → G such that for any connected finite étale subcovering G  → G of G  → G that factors through G † → G, the image of the composite ab/edge H ∩ Π G  → Π G   Π G  ab/edge — where we write Π G  for the torsion-free [cf. [CmbGC], Remark 1.1.4] quotient of the abelianization Π ab G  by the closed subgroup topologically generated by the images in Π ab G  of the edge-like subgroups of Π G  — is trivial. 22 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. The implications (i) ⇒ (ii) ⇒ (iv) are immediate. The equiv- alence (iii) ⇔ (iv) follows immediately from [NodNon], Lemma 1.6. Thus, to complete the verification of Proposition 1.4, it suffices to ver- ify the implication (iii) ⇒ (i). To this end, suppose that condition (iii) holds. First, we observe that, to verify the implication (iii) ⇒ (i), it suffices to verify the following assertion: Claim 1.4.A: Let γ 1 , γ 2 ∈ H be nontrivial elements.  for the unique elements of Write e  1 , e  2 ∈ Edge( G)  such that γ 1 ∈ Π e  1 , γ 2 ∈ Π e  2 [cf. Remark 1.1.1]. Edge( G) Then e  1 = e  2 . To verify Claim 1.4.A, let us observe that it follows from condition (iii) that, for every positive integer n, it holds that γ 1 n γ 2 n is edge-like, hence verticial. Thus, it follows immediately from Lemma 1.3 that there  such that { v ); in particular, exists an element v  ∈ Vert( G) e 1 , e  2 } ⊆ E( it holds that γ 1 , γ 2 ∈ Π v  . Thus, to complete the verification of Claim 1.4.A, we may assume without loss of generality — by replacing Π G , H  by Π v  , Π v  ∩ H, respectively — that Node(G) = ∅ [so e  1 , e  2 ∈ Cusp( G)]. Moreover, we may assume without loss of generality — by replacing Π G (respectively, γ 1 , γ 2 ) by a suitable open subgroup of Π G (respectively, suitable powers of γ 1 , γ 2 ) — that #Cusp(G) ≥ 4. Thus, it follows immediately from the well-known structure of the abelianization of the fundamental group of a smooth curve over an algebraically closed field of characteristic ∈ Σ that the direct product of any 3 cuspidal inertia subgroups of Π G associated to distinct cusps of G maps injectively to the abelianization Π ab G of Π G . In particular, since γ 1 γ 2 is edge-like, hence cuspidal, we conclude, by considering the cuspidal inertia subgroups that contain γ 1 , γ 2 , and γ 1 γ 2 , that e  1 = e  2 . This completes the proof of Claim 1.4.A, hence also of the implication (iii) ⇒ (i). This completes the proof of Proposition 1.4.  Proposition 1.5 (Group-theoretic characterization of closed subgroups of verticial subgroups). Let H ⊆ Π G be a closed sub- group of Π G . Then the following conditions are equivalent: (i) H is contained in a verticial subgroup. (ii) An open subgroup of H is contained in a verticial subgroup. (iii) Every element of H is verticial [cf. Definition 1.1]. (iv) There exists a connected finite étale subcovering G † → G of G  → G such that for any connected finite étale subcovering G  → G of G  → G that factors through G † → G, the image of the composite H ∩ Π G  → Π G   Π ab-comb G  COMBINATORIAL ANABELIAN TOPICS II 23 — where we write Π ab-comb for the torsion-free [cf. [CmbGC], G  Remark 1.1.4] quotient of the abelianization Π ab G  by the closed subgroup topologically generated by the images in Π ab G  of the verticial subgroups of Π G  — is trivial. Proof. The implications (i) ⇒ (ii) ⇒ (iv) are immediate. Next, we verify the implication (iv) ⇒ (iii). Suppose that condition (iv) holds. Let γ ∈ H. Then to verify that γ is verticial, we may assume with- out loss of generality — by replacing H by the procyclic subgroup of H topologically generated by γ — that H is procyclic. Now the implication (iv) ⇒ (iii) follows immediately from a similar argument to the argument applied in the proof of the implication (ii) ⇒ (i) of [NodNon], Lemma 1.6, in the edge-like case. Here, we note that unlike the edge-like case, there is a slight complication arising from the fact  is not [cf. [NodNon], Lemma 1.9, (i)] that an element v  ∈ Vert( G) necessarily uniquely determined by the condition that H ⊆ Π v  , i.e.,  such that e ) for some e  ∈ Node( G) there may exist distinct v  1 , v  2 ∈ V( H ⊆ Π e  = Π v  1 ∩ Π v  2 . On the other hand, this phenomenon is, in fact, irrelevant to the argument in question, since Π G does not contain any elements that fix, but permute the branches of, e  . This completes the proof of the implication (iv) ⇒ (iii). Finally, we verify the implication (iii) ⇒ (i). Suppose that condition (iii) holds. Now if every element of H is edge-like, then the implication (iii) ⇒ (i) follows from the implication (iii) ⇒ (i) of Proposition 1.4, together with the fact that every edge-like subgroup is contained in a verticial subgroup. Thus, to verify the implication (iii) ⇒ (i), we may assume without loss of generality that there exists an element γ 1 ∈ H  for the unique element of H that is not edge-like. Write v  1 ∈ Vert( G)  such that γ 1 ∈ Π v  1 [cf. Remark 1.1.1]. of Vert( G) Now we claim the following assertion: Claim 1.5.A: H ⊆ Π v  1 . Indeed, let γ 2 ∈ H be a nontrivial element of H. If γ 2 = γ 1 , then γ 2 ∈ Π v  1 . Thus, we may assume without loss of generality that γ 1  = γ 2 . def Write γ = γ 1 γ 2 −1 .  for the Next, suppose that γ 2 is not edge-like. Write v  2 ∈ Vert( G)  such that γ 2 ∈ Π v  2 [cf. Remark 1.1.1]. Let unique element of Vert( G) H → G be a connected finite étale subcovering of G  → G. Then since neither γ 1 nor γ 2 is edge-like, one verifies easily — by applying the implication (iv) ⇒ (i) of Proposition 1.4 to the closed subgroups of Π G topologically generated by γ 1 , γ 2 , respectively — that there exist a connected finite étale subcovering H  → H of G  → H and a positive integer n such that γ 1 n , γ 2 n ∈ Π H  ⊆ Π H , and, moreover, the images ab/edge [cf. the of γ 1 n , γ 2 n ∈ Π H  via the natural surjection Π H   Π H  notation of Proposition 1.4, (iv)] are nontrivial. Thus, it follows from 24 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI the existence of the natural split injection ab/edge Π v ab/edge −→ Π H  v∈Vert(H  ) of [NodNon], Lemma 1.4, together with the fact that γ 1 n γ 2 n ∈ Π H  is verticial [cf. condition (iii)], that v  1 (H  ) = v  2 (H  ), hence that v  1 (H) = v  2 (H). Therefore, by allowing the subcovering H → G of G  → G to vary, we conclude that v  1 = v  2 ; in particular, it holds that γ 2 ∈ Π v  1 . Next, suppose that γ 2 is edge-like, but that γ is not edge-like. Then, by applying the argument of the preceding paragraph concerning γ 2 to γ, we conclude that γ, hence also γ 2 , is contained in Π v  1 . Next, suppose that both γ 2 and γ are edge-like. Write e  2 , e  ∈  for the unique elements of Edge( G)  such that γ 2 ∈ Π e  2 , γ ∈ Π e  Edge( G) [cf. Remark 1.1.1]. Then since γ 1 is not edge-like, it follows immedi- ately that e  2  = e  . Moreover, it follows from condition (iii) that for any positive integer n, the element γ 2 n γ n is verticial. Thus, it follows  immediately from Lemma 1.3 that there exists a unique v  ∈ Vert( G) such that { e 2 , e  } ⊆ E( v ), γ 1 = γγ 2 ∈ Π v  . On the other hand, since  is uniquely determined by the condition that γ 1 ∈ Π v  1 , we v  1 ∈ Vert( G) thus conclude that v  1 = v  , hence that γ 2 ∈ Π e  2 ⊆ Π v  1 , as desired. This completes the proof of Claim 1.5.A and hence also of the implication (iii) ⇒ (i).  Theorem 1.6 (Section conjecture-type result for outer rep- resentations of SNN-, IPSC-type). Let Σ be a nonempty set of prime numbers, G a semi-graph of anabelioids of pro-Σ PSC-type, and I → Aut(G) an outer representation of SNN-type [cf. [NodNon], Definition 2.4, (iii)]. Write Π G for the [pro-Σ] fundamental group of G def out and Π I = Π G  I [cf. the discussion entitled “Topological groups” in [CbTpI], §0]; thus, we have a natural exact sequence of profinite groups 1 −→ Π G −→ Π I −→ I −→ 1 . Write Sect(Π I /I) for the set of sections of the natural surjection Π I  I. Then the following hold:  the composite I v  → Π I  I [cf. [NodNon], (i) For any v  ∈ Vert( G), Definition 2.2, (i)] is an isomorphism. In particular, I v  ⊆ Π I determines an element s v  ∈ Sect(Π I /I); thus, we have a map  −→ Sect(Π I /I) Vert( G) v  → s v  . Finally, the following equalities concerning centralizers of sub- groups of Π I in Π G [cf. the discussion entitled “Topological COMBINATORIAL ANABELIAN TOPICS II 25 groups” in “Notations and Conventions”] hold: Z Π G (s v  (I)) = Z Π G (I v  ) = Π v  . (ii) The map of (i) is injective. (iii) If, moreover, I → Aut(G) is of IPSC-type [cf. [NodNon], Definition 2.4, (i)], then, for any s ∈ Sect(Π I /I), the central- izer Z Π G (s(I)) is contained in a verticial subgroup. (iv) Let s ∈ Sect(Π I /I). Consider the following two conditions: (1) The section s is contained in the image of the map of (i),  i.e., s = s v  for some v  ∈ Vert( G). (2) Z Π G (Z Π G (s(I))) = {1}. Then we have an implication (1) =⇒ (2) . If, moreover, I → Aut(G) is of IPSC-type, then we have an equivalence (1) ⇐⇒ (2) . Proof. First, we verify assertion (i). The fact that the composite I v  → Π I  I is an isomorphism follows from condition (2  ) of [NodNon], Definition 2.4, (ii). On the other hand, the equalities Z Π G (s v  (I)) = Z Π G (I v  ) = Π v  follow from [NodNon], Lemma 3.6, (i). This completes the proof of assertion (i). Assertion (ii) follows immediately from the final equalities of assertion (i), together with [NodNon], Lemma 1.9, def (ii). Next, we verify assertion (iii). Write H = Z Π G (s(I)). Then it follows immediately from [CmbGC], Proposition 2.6, together with the definition of H = Z Π G (s(I)), that for any connected finite étale subcovering G  → G of G  → G, the image of the composite H ∩ Π G  → Π G   Π ab-comb G  [cf. the notation of Proposition 1.5, (iv)] is trivial. Thus, it follows from the implication (iv) ⇒ (i) of Proposition 1.5 that H is contained in a verticial subgroup. This completes the proof of assertion (iii). Finally, we verify assertion (iv). To verify the implication (1) ⇒ (2), suppose that condition (1) holds. Then since Z Π G (s v  (I)) = Z Π G (I v  ) = Π v  [cf. assertion (i)] is commensurably terminal in Π G [cf. [CmbGC], Proposition 1.2, (ii)] and center-free [cf. [CmbGC], Remark 1.1.3], we conclude that Z Π G (Z Π G (s v  (I))) = Z Π G (Π v  ) = {1}. This completes the proof of the implication (1) ⇒ (2). Next, suppose that I → Aut(G) is of IPSC-type, and that condition (2) holds. Then it follows from assertion  such that H def = Z Π G (s(I)) ⊆ (iii) that there exists a v  ∈ Vert( G) Π v  , so I v  ⊆ Z Π I (H). On the other hand, since s(I) ⊆ Z Π I (H), and Z Π G (H) = Z Π G (Z Π G (s(I))) = {1} [cf. condition (2)], i.e., the composite of natural homomorphisms Z Π I (H) → Π I  I is injective, it follows 26 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI that s(I) = Z Π I (H) ⊇ I v  . Since I v  and s(I) may be obtained as the images of sections, we thus conclude that I v  = s(I), i.e., s = s v  . This completes the proof of the implication (2) ⇒ (1), hence also of assertion (iv).  Remark 1.6.1. Recall that in the case of outer representations of NN- type, the period matrix is not necessarily nondegenerate [cf. [CbTpI], Remark 5.9.2]. In particular, the argument applied in the proof of The- orem 1.6, (iii) — which depends, in an essential way, on the fact that, in the case of outer representations of IPSC-type, the period matrix is nondegenerate [cf. the proof of [CmbGC], Proposition 2.6] — cannot be applied in the case of outer representations of NN-type. Nevertheless, the question of whether or not Theorem 1.6, (iii), as well as the appli- cation of Theorem 1.6, (iii), given in Corollary 1.7, (ii), below, may be generalized to the case of outer representations of NN-type remains a topic of interest to the authors. Corollary 1.7 (Group-theoretic characterization of verticial subgroups for outer representations of IPSC-type). In the no- tation of Theorem 1.6, let us refer to a closed subgroup of Π G as a section-centralizer if it may be written in the form Z Π G (s(I)) for some s ∈ Sect(Π I /I). Let H ⊆ Π G be a closed subgroup of Π G . Then the following hold: (i) Suppose that H is a section-centralizer such that Z Π G (H) = {1}. Then the following conditions on a section s ∈ Sect(Π I /I) are equivalent: (i-1) H = Z Π G (s(I)). (i-2) s(I) ⊆ Z Π I (H). (i-3) s(I) = Z Π I (H). (ii) Consider the following three conditions: (ii-1) H is a verticial subgroup. (ii-2) H is a section-centralizer such that Z Π G (H) = {1}. (ii-3) H is a maximal section-centralizer. Then we have implications (ii-1) =⇒ (ii-2) =⇒ (ii-3) . If, moreover, I → Aut(G) is of IPSC-type [cf. [NodNon], Definition 2.4, (i)], then we have equivalences (ii-1) ⇐⇒ (ii-2) ⇐⇒ (ii-3) . COMBINATORIAL ANABELIAN TOPICS II 27 Proof. First, we verify assertion (i). The implication (i-1) ⇒ (i-2) is immediate. To verify the implication (i-2) ⇒ (i-3), suppose that condition (i-2) holds. Then since Z Π I (H) ∩ Π G = Z Π G (H) = {1}, the composite Z Π I (H) → Π I  I is injective. Thus, since the composite s(I) → Z Π I (H) → Π I  I is an isomorphism, it follows immediately that condition (i-3) holds. This completes the proof of the implication (i-2) ⇒ (i-3). Finally, to verify the implication (i-3) ⇒ (i-1), suppose that condition (i-3) holds. Then since H is a section-centralizer, there exists a t ∈ Sect(Π I /I) such that H = Z Π G (t(I)). In particular, t(I) ⊆ Z Π I (H) = s(I) [cf. condition (i-3)]. We thus conclude that t = s, i.e., that condition (i-1) holds. This completes the proof of assertion (i). Next, we verify assertion (ii). The implication (ii-1) ⇒ (ii-2) fol- lows immediately from Theorem 1.6, (i), (iv). To verify the impli- cation (ii-2) ⇒ (ii-3), suppose that H satisfies condition (ii-2); let s ∈ Sect(Π I /I) be such that H ⊆ Z Π G (s(I)). Then it follows imme- diately that s(I) ⊆ Z Π I (H). Thus, it follows immediately from the equivalence (i-1) ⇔ (i-2) of assertion (i) that H = Z Π G (s(I)). This completes the proof of the implication (ii-2) ⇒ (ii-3). Finally, observe that the implication (ii-3) ⇒ (ii-1) in the case where I → Aut(G) is of IPSC-type follows immediately from Theorem 1.6, (iii), together with the fact that every verticial subgroup is a section-centralizer [cf. the implication (ii-1) ⇒ (ii-2) verified above]. This completes the proof of Corollary 1.7.  Lemma 1.8 (Group-theoretic characterization of verticial sub- groups for outer representations of SNN-type). Let H ⊆ Π G be a closed subgroup of Π G and I → Aut(G) an outer representation of def out SNN-type [cf. [NodNon], Definition 2.4, (iii)]. Write Π I = Π G  I [cf. the discussion entitled “Topological groups” in [CbTpI], §0]; thus, we have a natural exact sequence of profinite groups 1 −→ Π G −→ Π I −→ I −→ 1 . Suppose that G is untangled [cf. [NodNon], Definition 1.2]. Then H is a verticial subgroup if and only if H satisfies the following four conditions: def (i) The composite I H = Z Π I (H) → Π I  I is an isomorphism. (ii) It holds that H = Z Π G (I H ). (iii) For any γ ∈ Π G , it holds that γ ∈ H if and only if H ∩ (γ · H · γ −1 )  = {1}. (iv) H contains a nontrivial verticial element of Π G [cf. Defini- tion 1.1]. 28 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. If H is a verticial subgroup, then it is immediate that condition (iv) is satisfied; moreover, it follows from condition (2  ) of [NodNon], Definition 2.4, (ii) (respectively, [NodNon], Lemma 3.6, (i); [NodNon], Remark 1.10.1), that H satisfies condition (i) (respectively, (ii); (iii)). This completes the proof of necessity. To verify sufficiency, suppose that H satisfies conditions (i), (ii), (iii),  and (iv). It follows from condition (iv) that there exists a v  ∈ Vert( G) def such that J = H ∩ Π v   = {1}. If either J = Π v  or J = H, i.e., either Π v  ⊆ H or H ⊆ Π v  , then it is immediate that either I H ⊆ I v  or I v  ⊆ I H [cf. [NodNon], Definition 2.2, (i)]. Thus, it follows from condition (i) [for H and Π v  ] that I H = I v  . But then it follows from condition (ii) [for H and Π v  ] that H = Z Π G (I H ) = Z Π G (I v  ) = Π v  ; in particular, H is a verticial subgroup. Thus, we may assume without loss of generality that J  = H, Π v  . def Let γ ∈ H \ J. Write J γ = γ · J · γ −1 . Then we have inclusions Π v  ⊇ J ⊆ H ⊇ J γ ⊆ Π v  γ (= γ · Π v  · γ −1 ) . Now we claim the following assertion: Claim 1.8.A: N Π G (J) = J, N Π G (J γ ) = J γ . Indeed, let σ ∈ N Π G (J). Then since {1}  = J = J ∩ (σ · J · σ −1 ) ⊆ Π v  ∩ Π v  σ , it follows from condition (iii) [for Π v  ] that σ ∈ Π v  . Similarly, since {1}  = J = J ∩ (σ · J · σ −1 ) ⊆ H ∩ (σ · H · σ −1 ), it follows from condition (iii) [for H] that σ ∈ H. Thus, σ ∈ Π v  ∩ H = J. In particular, we obtain that N Π G (J) = J. A similar argument implies that N Π G (J γ ) = J γ . This completes the proof of Claim 1.8.A. Now the composites N Π I (J), N Π I (J γ ) → Π I  I fit into exact sequences of profinite groups 1 −→ N Π G (J) −→ N Π I (J) −→ I , 1 −→ N Π G (J γ ) −→ N Π I (J γ ) −→ I . Thus, since we have inclusions I H = Z Π I (H) ⊆ Z Π I (J) ⊆ N Π I (J) , I H = Z Π I (H) ⊆ Z Π I (J γ ) ⊆ N Π I (J γ ) , I v  = Z Π I (Π v  ) ⊆ Z Π I (J) ⊆ N Π I (J) , I v  γ = Z Π I (Π v  γ ) ⊆ Z Π I (J γ ) ⊆ N Π I (J γ ) , it follows immediately from Claim 1.8.A, together with condition (i) [for H and Π v  ], that N Π I (J) = J · I H = J · I v  , N Π I (J γ ) = J γ · I H = J γ · I v  γ . In particular, we obtain that I H ⊆ N Π I (J) = J · I v  ⊆ Π v  · D v  = D v  , I H ⊆ N Π I (J γ ) = J γ · I v  γ ⊆ Π v  γ · D v  γ = D v  γ COMBINATORIAL ANABELIAN TOPICS II 29 [cf. [NodNon], Definition 2.2, (i)], i.e., I H ⊆ D v  ∩ D v  γ . On the other hand, since H  γ ∈ J = H ∩ Π v  , it follows from condition (iii) [for Π v  ] that Π v  γ ∩ Π v  = {1}; thus, it follows immediately from the fact that D v  ∩ D v  γ ∩ Π G = Π v  ∩ Π v  γ = {1} [cf. [CmbGC], Proposition 1.2, (ii)], together with condition (i), that I H = D v  ∩ D v  γ , which implies,  by [NodNon], Proposition 3.9, (iii), that there exists a w  ∈ Vert( G) such that I H = I w  . In particular, it follows from condition (ii) [for H and Π w  ] that H = Z Π G (I H ) = Z Π G (I w  ) = Π w  . Thus, H is a verticial subgroup. This completes the proof of Lemma 1.8.  Theorem 1.9 (Group-theoretic verticiality/nodality of isomor- phisms of outer representations of NN-, IPSC-type). Let Σ be a nonempty set of prime numbers, G (respectively, H) a semi-graph of anabelioids of pro-Σ PSC-type, Π G (respectively, Π H ) the [pro-Σ] ∼ fundamental group of G (respectively, H), α : Π G → Π H an isomor- phism of profinite groups, I (respectively, J) a profinite group, ρ I : I → Aut(G) (respectively, ρ J : J → Aut(H)) a continuous homomorphism, ∼ and β : I → J an isomorphism of profinite groups. Suppose that the diagram I −−−→ Out(Π G ) ⏐ ⏐ ⏐ ⏐ Out(α) β   J −−−→ Out(Π H ) — where the right-hand vertical arrow is the isomorphism induced by α; the upper and lower horizontal arrows are the homomorphisms de- termined by ρ I and ρ J , respectively — commutes. Then the following hold: (i) Suppose, moreover, that ρ I , ρ J are of NN-type [cf. [NodNon], Definition 2.4, (iii)]. Then the following three conditions are equivalent: (1) The isomorphism α is group-theoretically verticial [i.e., roughly speaking, preserves verticial subgroups — cf. [CmbGC], Definition 1.4, (iv)]. (2) The isomorphism α is group-theoretically nodal [i.e., roughly speaking, preserves nodal subgroups — cf. [NodNon], Definition 1.12]. (3) There exists a nontrivial verticial element γ ∈ Π G such that α(γ) ∈ Π H is verticial [cf. Definition 1.1]. (ii) Suppose, moreover, that ρ I is of NN-type, and that ρ J is of IPSC-type [cf. [NodNon], Definition 2.4, (i)]. [For example, this will be the case if both ρ I and ρ J are of IPSC-type — cf. 30 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [NodNon], Remark 2.4.2.] Then α is group-theoretically verticial, hence also [cf. (i)] group-theoretically nodal. Proof. First, we verify assertion (i). The implication (1) ⇒ (2) fol- lows from [NodNon], Proposition 1.13. The implication (2) ⇒ (3) follows from the fact that any nodal subgroup is contained in a verti- cial subgroup. [Note that if Node(H) = ∅, then every element of Π H is verticial.] Finally, we verify the implication (3) ⇒ (1). Suppose that condition (3) holds. Since verticial subgroups are commensurably terminal [cf. [CmbGC], Proposition 1.2, (ii)], to verify the implica- tion (3) ⇒ (1), by replacing Π I , Π J by open subgroups of Π I , Π J , we may assume without loss of generality that ρ I , ρ J are of SNN-type [cf. [NodNon], Definition 2.4, (iii)], and, moreover, that G and H are un- tangled [cf. [NodNon], Definition 1.2; [NodNon], Remark 1.2.1, (i), (ii)].  be such that γ ∈ Π v  . Then it is immediate that α(Π v  ) Let v  ∈ Vert( G) satisfies conditions (i), (ii), and (iii) in the statement of Lemma 1.8. On the other hand, it follows from condition (3) that α(Π v  ) satisfies condition (iv) in the statement of Lemma 1.8. Thus, it follows from Lemma 1.8 that α(Π v  ) ⊆ Π H is a verticial subgroup. Now it follows from [NodNon], Theorem 4.1, that α is group-theoretically verticial. This completes the proof of the implication (3) ⇒ (1). Finally, we verify assertion (ii). It is immediate that, to verify as- sertion (ii) — by replacing I, J by open subgroups of I, J — we may assume without loss of generality that ρ I is of SNN-type. Let H ⊆ Π G be a verticial subgroup of Π G . Then it follows from Corollary 1.7, (ii), that H, hence also α(H), is a maximal section-centralizer [cf. the statement of Corollary 1.7]. Thus, since ρ J is of IPSC-type, again by Corollary 1.7, (ii), we conclude that α(H) ⊆ Π H is a verticial subgroup of Π H . In particular, it follows from [NodNon], Theorem 4.1, together with [NodNon], Remark 2.4.2, that α is group-theoretically verticial and group-theoretically nodal. This completes the proof of assertion (ii).  Remark 1.9.1. Thus, Theorem 1.9, (i), may be regarded as a gen- eralization of [NodNon], Corollary 4.2. Of course, ideally, one would like to be able to prove that conditions (1) and (2) of Theorem 1.9, (i), hold automatically [i.e., as in the case of outer representations of IPSC-type treated in Theorem 1.9, (ii)], without assuming condition (3). Although this topic lies beyond the scope of the present mono- graph, perhaps progress could be made in this direction if, say, in the case where Σ is either equal to the set of all prime numbers or of car- dinality one, one starts with an isomorphism α that arises from a PF- admissible [cf. [CbTpI], Definition 1.4, (i)] isomorphism between con- figuration space groups corresponding to m-dimensional configuration spaces [where m ≥ 2] associated to stable curves that give rise to G and COMBINATORIAL ANABELIAN TOPICS II 31 H, respectively [i.e., one assumes the condition of “m-cuspidalizability” discussed in Definition 3.20, below, where we replace the condition of “PFC-admissibility” by the condition of “PF-admissibility”]. For in- stance, if Cusp(G)  = ∅, then it follows from [CbTpI], Theorem 1.8, (iv); [NodNon], Corollary 4.2, that this condition on α is sufficient to imply that conditions (1) and (2) of Theorem 1.9, (i), hold. 32 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 2. Partial combinatorial cuspidalization for F-admissible outomorphisms In the present §2, we apply the results obtained in the preceding §1, together with the theory developed by the authors in earlier papers, to prove combinatorial cuspidalization-type results for F-admissible out- omorphisms [cf. Theorem 2.3, (i), below]. We also show that any F- admissible outomorphism of a configuration space group [arising from a configuration space] of sufficiently high dimension [i.e., ≥ 3 in the affine case; ≥ 4 in the proper case] is necessarily C-admissible, i.e., preserves the cuspidal inertia subgroups of the various subquotients correspond- ing to surface groups [cf. Theorem 2.3, (ii), below]. Finally, we discuss applications of these combinatorial anabelian results to the anabelian geometry of configuration spaces associated to hyperbolic curves over arithmetic fields [cf. Corollaries 2.5, 2.6, below]. In the present §2, let Σ be a set of prime numbers which is either equal to the set of all prime numbers or of cardinality one; n a positive integer; k an algebraically closed field of characteristic ∈ Σ; X a hyper- bolic curve of type (g, r) over k. For each positive integer i, write X i for the i-th configuration space of X; Π i for the maximal pro-Σ quotient of the fundamental group of X i . Definition 2.1. Let α ∈ Aut(Π n ) be an automorphism of Π n . (i) Write {1} = K n ⊆ K n−1 ⊆ · · · ⊆ K 2 ⊆ K 1 ⊆ K 0 = Π n for the standard fiber filtration on Π n [cf. [CmbCsp], Defini- tion 1.1, (i)]. For each m ∈ {1, 2, · · · , n}, write C m for the [finite] set of K m−1 /K m -conjugacy classes of cuspidal inertia subgroups of K m−1 /K m [where we recall that K m−1 /K m is equipped with a natural structure of pro-Σ surface group — cf. [MzTa], Definition 1.2]. Then we shall say that α is wC- admissible [i.e., “weakly C-admissible”] if α preserves the stan- dard fiber filtration on Π n and, moreover, satisfies the following conditions: • If m ∈ {1, 2, · · · n−1}, then the automorphism of K m−1 /K m determined by α induces an automorphism of C m . • It follows immediately from the various definitions in- volved that we have a natural injection C n−1 → C n . That is to say, if one thinks of K n−2 as the two-dimensional configuration space group associated to some hyperbolic curve, then the image of C n−1 → C n corresponds to the set of cusps of a fiber [of the two-dimensional configura- tion space over the hyperbolic curve] that arise from the COMBINATORIAL ANABELIAN TOPICS II 33 cusps of the hyperbolic curve. Then the automorphism of K n−1 determined by α induces an automorphism of the image of the natural injection C n−1 → C n . Write Aut wC (Π n ) ⊆ Aut(Π n ) for the subgroup of wC-admissible automorphisms and def Out wC (Π n ) = Aut wC (Π n )/Inn(Π n ) ⊆ Out(Π n ) . We shall refer to an element of Out wC (Π n ) as a wC-admissible outomorphism. (ii) We shall say that α is FwC-admissible if α is F-admissible [cf. [CmbCsp], Definition 1.1, (ii)] and wC-admissible [cf. (i)]. Write Aut FwC (Π n ) ⊆ Aut F (Π n ) for the subgroup of FwC-admissible automorphisms and def Out FwC (Π n ) = Aut FwC (Π n )/Inn(Π n ) ⊆ Out F (Π n ) . We shall refer to an element of Out FwC (Π n ) as an FwC-admissible outomorphism. (iii) We shall say that α is DF-admissible [i.e., “diagonal-fiber- admissible”] if α is F-admissible, and, moreover, α induces the same automorphism of Π 1 relative to the various quotients Π n  Π 1 by fiber subgroups of co-length 1 [cf. [MzTa], Defini- tion 2.3, (iii)]. Write Aut DF (Π n ) ⊆ Aut F (Π n ) for the subgroup of DF-admissible automorphisms. Remark 2.1.1. Thus, it follows immediately from the definitions that C-admissible =⇒ wC-admissible. In particular, we have inclusions Aut FC (Π n ) ⊂ Aut FwC (Π n ) ∩ Aut C (Π n ) ⊂ Out FC (Π n ) ⊂ Out FwC (Π n ) ∩ ∩ Aut wC (Π n ) Out C (Π n ) [cf. Definition 2.1, (i), (ii)]. ∩ ⊂ Out wC (Π n ) 34 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Lemma 2.2 (F-admissible automorphisms and inertia subgroups). Let α ∈ Aut F (Π n ) be an F-admissible automorphism of Π n . Then the following hold: (i) There exist β ∈ Aut DF (Π n ) [cf. Definition 2.1, (iii)] and ι ∈ Inn(Π n ) such that α = β ◦ ι. (ii) For each positive integer i, write Z i log for the i-th log config- uration space of X [cf. the discussion entitled “Curves” in “Notations and Conventions”]; U Z i ⊆ Z i for the interior of Z i log [cf. the discussion entitled “Log schemes” in [CbTpI], §0], which may be identified with X i . Let be an irreducible component of the complement Z n−1 \ U Z n−1 [cf. [CmbCsp], Proposition 1.3]; I  ⊆ Π n−1 an inertia subgroup of Π n−1 asso- ciated to the divisor of Z n−1 ; pr : U Z n → U Z n−1 the projection obtained by forgetting the factor labeled n; pr Π : Π n  Π n−1 def the surjection induced by pr; Π n/n−1 = Ker(pr Π ); θ an ir- reducible component of the fiber of the [uniquely determined] extension Z n → Z n−1 of pr over the generic point of [so θ naturally determines an irreducible component of the com- plement Z n \ U Z n ]; D I θ ⊆ Π n × Π n−1 I  (⊆ Π n ) — where the homomorphism Π n → Π n−1 implicit in the fiber product is the surjection pr Π : Π n  Π n−1 — a decomposition subgroup of Π n × Π n−1 I  (⊆ Π n ) associated to the divisor [naturally deter- def mined by] θ of Z n ; Π θ = D I θ ∩ Π n/n−1 [cf. [CmbCsp], Proposi- tion 1.3, (iv)]. Suppose that the automorphism of Π n−1 induced by α ∈ Aut F (Π n ) relative to pr Π stabilizes I  ⊆ Π n−1 . Then α preserves the Π n/n−1 -conjugacy class of Π θ . Proof. Assertion (i) follows immediately from [CbTpI], Theorem A, (i). Assertion (ii) follows immediately from Theorem 1.9, (ii) [cf. also the proof of [CmbCsp], Proposition 1.3, (iv)].  Theorem 2.3 (Partial combinatorial cuspidalization for F-ad- missible outomorphisms). Let Σ be a set of prime numbers which is either equal to the set of all prime numbers or of cardinality one; n a positive integer; X a hyperbolic curve of type (g, r) over an alge- braically closed field of characteristic ∈ Σ; X n the n-th configuration space of X; Π n the maximal pro-Σ quotient of the fundamental group of X n ; Out F (Π n ) ⊆ Out(Π n ) the subgroup of F-admissible outomorphisms [i.e., roughly speaking, outomorphisms that preserve the fiber subgroups — cf. [CmbCsp], Def- inition 1.1, (ii)] of Π n ; Out FC (Π n ) ⊆ Out F (Π n ) COMBINATORIAL ANABELIAN TOPICS II 35 the subgroup of FC-admissible outomorphisms [i.e., roughly speak- ing, outomorphisms that preserve the fiber subgroups and the cuspidal inertia subgroups — cf. [CmbCsp], Definition 1.1, (ii)] of Π n ; (Out FC (Π n ) ⊆) Out FwC (Π n ) ⊆ Out F (Π n ) the subgroup of FwC-admissible outomorphisms [cf. Definition 2.1, (ii); Remark 2.1.1] of Π n . Then the following hold: (i) Write  def n inj = 1 2 if r  = 0, if r = 0 ,  def n bij = 3 4 if r  = 0, if r = 0 . If n ≥ n inj (respectively, n ≥ n bij ), then the natural homomor- phism Out F (Π n+1 ) −→ Out F (Π n ) induced by the projections X n+1 → X n obtained by forgetting any one of the n + 1 factors of X n+1 [cf. [CbTpI], Theorem A, (i)] is injective (respectively, bijective). (ii) Write ⎧ ⎨ 2 def 3 n FC = ⎩ 4 if (g, r) = (0, 3), if (g, r)  = (0, 3) and r  = 0, if r = 0 . If n ≥ n FC , then it holds that Out FC (Π n ) = Out F (Π n ) . (iii) Write ⎧ ⎨ 2 def 3 n FwC = ⎩ 4 if r ≥ 2, if r = 1, if r = 0 . If n ≥ n FwC , then it holds that Out FwC (Π n ) = Out F (Π n ) . (iv) Consider the natural inclusion S n → Out(Π n ) — where we write S n for the symmetric group on n letters — obtained by permuting the various factors of X n . If (r, n)  = (0, 2), then the image of this inclusion is contained in the cen- tralizer Z Out(Π n ) (Out F (Π n )). Proof. First, we verify assertion (iii) in the case where n = 2, which implies that r ≥ 2 [cf. the statement of assertion (iii)]. To verify 36 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI assertion (iii) in the case where n = 2, it is immediate that it suffices to verify that Aut FwC (Π 2 ) = Aut F (Π 2 ) . Let α ∈ Aut F (Π 2 ). Let us assign the cusps of X the labels a 1 , · · · , a r . Now, for each i ∈ {1, · · · , r}, recall that there is a uniquely determined cusp of the geometric generic fiber X 2/1 of the projection X 2 → X to the factor labeled 1 that corresponds naturally to the cusp of X labeled a i ; we assign to this uniquely determined cusp the label b i . Thus, there is precisely one cusp of X 2/1 that has not been assigned a label ∈ {b 1 , · · · , b r }; we assign to this uniquely determined cusp the label b r+1 . Then since the automorphism of Π 1 induced by α relative to either p 1 or p 2 — where we write p 1 , p 2 for the surjections Π 2  Π 1 induced by the projections X 2 → X to the factors labeled 1, 2, respectively — is FC-admissible [cf. [CbTpI], Theorem A, (ii)], it follows from the various definitions involved that, to verify that α ∈ Aut FwC (Π 2 ), it suffices to verify the following assertion: def Claim 2.3.A: For any b ∈ {b 1 , · · · , b r }, if I b ⊆ Π 2/1 = Ker(p 1 ) ⊆ Π 2 is a cuspidal inertia subgroup associated to the cusp labeled b, then α(I b ) is a cuspidal inertia subgroup. Now observe that to verify Claim 2.3.A, by replacing α by the compos- ite of α with a suitable element of Aut FC (Π 2 ) [cf. [CmbCsp], Lemma 2.4], we may assume without loss of generality that the [necessarily FC- admissible] automorphism of Π 1 induced by α relative to p 1 , hence also relative to p 2 [cf. [CbTpI], Theorem A, (i)], induces the identity auto- morphism on the set of conjugacy classes of cuspidal inertia subgroups of Π 1 . To verify Claim 2.3.A, let us fix b ∈ {b 1 , · · · , b r }, together with a cuspidal inertia subgroup I b ⊆ Π 2/1 associated to the cusp labeled b of Π 2/1 . Also, let us fix • a ∈ {a 1 , · · · , a r } such that if b = b i and a = a j , then i  = j [cf. the assumption that r ≥ 2!]; • a cuspidal inertia subgroup I a ⊆ Π 1 associated to the cusp labeled a of Π 1 . Now observe that since the [necessarily FC-admissible] automorphism of Π 1 induced by α relative to p 1 induces the identity automorphism on the set of conjugacy classes of cuspidal inertia subgroups of Π 1 , to verify the fact that α(I b ) is a cuspidal inertia subgroup, we may assume without loss of generality [by replacing α by a suitable Π 2 -conjugate of α] that the automorphism of Π 1 induced by α relative to p 1 fixes I a . Let Π F a ⊆ Π 2/1 be a major verticial subgroup at a [cf. [CmbCsp], Definition 1.4, (ii)] such that I b ⊆ Π F a . Then it follows from Lemma 2.2, (ii), COMBINATORIAL ANABELIAN TOPICS II 37 that α fixes the Π 2/1 -conjugacy class of Π F a , i.e., that Π † F a = α(Π F a ) is a Π 2/1 -conjugate of Π F a . Thus, one verifies easily that, to verify that α(I b ) is a cuspidal inertia subgroup, it suffices to verify that the ∼ isomorphism Π F a → Π † F a induced by α is group-theoretically cuspidal — cf. [CmbGC], Definition 1.4, (iv). [Note that it follows immediately from the various definitions involved that Π F a and Π † F a may be regarded as pro-Σ fundamental groups of semi-graphs of anabelioids of pro-Σ PSC-type.] On the other hand, it follows immediately from the various definitions involved that this isomorphism factors as the composite def ∼ ∼ ∼ Π F a −→ Π 1 −→ Π 1 ←− Π † F a — where the first and third arrows are the isomorphisms induced by p 2 : Π 2  Π 1 [cf. [CmbCsp], Definition 1.4, (ii)], and the second ar- row is the automorphism induced by α relative to p 2 — and that the three arrows appearing in this composite are group-theoretically cuspi- dal. Thus, we conclude that α(I b ) is a cuspidal inertia subgroup. This completes the proof of Claim 2.3.A, hence also of assertion (iii) in the case where n = 2. Next, we verify assertion (ii) in the case where (g, r, n) = (0, 3, 2). In the following, we shall use the notation “a i ” [for i = 1, 2, 3] and “b j ” [for j = 1, 2, 3, 4] introduced in the proof of assertion (iii) in the case where n = 2. Now, to verify assertion (ii) in the case where (g, r, n) = (0, 3, 2), it is immediate that it suffices to verify that Aut FC (Π 2 ) = Aut F (Π 2 ) . Let α ∈ Aut F (Π 2 ). Then let us observe that to verify that α ∈ Aut FC (Π 2 ), by replacing α by the composite of α with a suitable ele- ment of Aut FC (Π 2 ) [cf. [CmbCsp], Lemma 2.4], we may assume without loss of generality that the [necessarily FC-admissible — cf. [CbTpI], Theorem A, (ii)] automorphism of Π 1 induced by α relative to p 1 , hence also relative to p 2 [cf. [CbTpI], Theorem A, (i)] — where we write p 1 , p 2 for the surjections Π 2  Π 1 induced by the projections X 2 → X to the factors labeled 1, 2, respectively — induces the identity automor- phism on the set of conjugacy classes of cuspidal inertia subgroups of Π 1 . Now it follows from assertion (iii) in the case where n = 2 that α is FwC-admissible; thus, to verify the fact that α is FC-admissible, it suffices to verify the following assertion: def Claim 2.3.B: If I b 4 ⊆ Π 2/1 = Ker(p 1 ) ⊆ Π 2 is a cuspi- dal inertia subgroup associated to the cusp labeled b 4 , then α(I b 4 ) is a cuspidal inertia subgroup. On the other hand, as is well-known [cf. e.g., [CbTpI], Lemma 6.10, (ii)], there exists an automorphism of X 2 over X relative to the projec- tion to the factor labeled 1 which switches the cusps on the geometric 38 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI generic fiber X 2/1 labeled b 1 and b 4 . In particular, there exists an auto- morphism ι of Π 2 over Π 1 relative to p 1 which switches the respective Π 2/1 -conjugacy classes of cuspidal inertia subgroups associated to b 1 and b 4 . Write β = ι −1 ◦ α ◦ ι. Now let us verify that Claim 2.3.B follows from the following asser- tion: Claim 2.3.C: β ∈ Aut F (Π 2 ). Indeed, if Claim 2.3.C holds, then it follows from assertion (iii) in the case where n = 2 that, for any cuspidal inertia subgroup I b 1 ⊆ Π 2/1 associated to the cusp labeled b 1 , β(I b 1 ) is a cuspidal inertia subgroup. Thus, it follows immediately from our choice of ι that, for any cuspidal inertia subgroup I b 4 ⊆ Π 2/1 associated to the cusp labeled b 4 , α(I b 4 ) is a cuspidal inertia subgroup. This completes the proof of the assertion that Claim 2.3.C implies Claim 2.3.B. Finally, we verify Claim 2.3.C. Since α and ι, hence also β, preserve Π 2/1 ⊆ Π 2 , it follows immediately from [CmbCsp], Proposition 1.2, (i), that, to verify Claim 2.3.C, it suffices to verify that β preserves Ξ 2 ⊆ Π 2 [cf. [CmbCsp], Definition 1.1, (iii)], i.e., the normal closed subgroup of Π 2 topologically normally generated by a cuspidal inertia subgroup associated to b 4 . On the other hand, this follows immediately from the fact that α preserves the Π 2/1 -conjugacy class of cuspidal inertia subgroups associated to b 1 [cf. assertion (iii) in the case where n = 2], together with our choice of ι. This completes the proof of Claim 2.3.C, hence also of assertion (ii) in the case where (g, r, n) = (0, 3, 2). Next, we verify assertion (ii) in the case where (g, r, n)  = (0, 3, 2). Thus, n ≥ 3. Write Π † 3 (respectively, Π † 2 ; Π † 1 ) for the kernel of the surjection Π n  Π n−3 (respectively, Π n−1  Π n−3 ; Π n−2  Π n−3 ) induced by the projection obtained by forgetting the factor(s) labeled n, n − 1, n − 2 (respectively, n − 1, n − 2; n − 2). Here, if n = 3, then def we set Π n−3 = Π 0 = {1}. Then recall [cf., e.g., the proof of [CmbCsp], Theorem 4.1, (i)] that we have natural isomorphisms out out out Π n ≃ Π † 3  Π n−3 ; Π n−1 ≃ Π † 2  Π n−3 ; Π n−2 ≃ Π † 1  Π n−3 [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. Also, we recall [cf. [MzTa], Proposition 2.4, (i)] that one may interpret the surjections Π † 3  Π † 2  Π † 1 induced by the surjections Π n  Π n−1  Π n−2 as the surjections “Π 3  Π 2  Π 1 ” that arise from the projec- tions X 3 → X 2 → X in the case of an “X” of type (g, r + n − 3). Moreover, one verifies easily that this interpretation is compatible with the definition of the various “Out(−)’s” involved. Thus, since n FC = 4 if r = 0, the above natural isomorphisms, together with [CbTpI], The- orem A, (ii), allow one to reduce the equality in question to the case where n = 3 and r  = 0. COMBINATORIAL ANABELIAN TOPICS II 39 Now one verifies easily that, to verify the equality in question in the case where n = 3 and r  = 0, it is immediate that it suffices to verify that Aut FC (Π 3 ) = Aut F (Π 3 ) . Let α ∈ Aut F (Π 3 ). Then let us observe that to verify α ∈ Aut FC (Π 3 ), by replacing α by the composite of α with a suitable element of Aut FC (Π 3 ) [cf. [CmbCsp], Lemma 2.4], we may assume without loss of general- ity that the [necessarily FC-admissible — cf. [CbTpI], Theorem A, (ii)] automorphism of Π 1 induced by α relative to q 1 , hence also rel- ative to either q 2 or q 3 [cf. [CbTpI], Theorem A, (i)] — where we write q 1 , q 2 , q 3 for the surjections Π 3  Π 1 induced by the projections X 3 → X to the factors labeled 1, 2, 3, respectively — induces the iden- tity automorphism on the set of conjugacy classes of cuspidal inertia subgroups of Π 1 ; in particular, one verifies easily that the [necessarily FC-admissible — cf. [CbTpI], Theorem A, (ii)] automorphism of Π 2/1 — where we write p 1 : Π 2  Π 1 for the surjection induced by the pro- def jection X 2 → X to the factor labeled 1 and Π 2/1 = Ker(p 1 ) ⊆ Π 2 — induced by α induces the identity automorphism on the set of conjugacy classes of cuspidal inertia subgroups of Π 2/1 . Write X 2/1 (respectively, X 3/2 ; X 3/1 ) for the geometric generic fiber of the projection X 2 → X (respectively, X 3 → X 2 ; X 3 → X) to the factor(s) labeled 1 (respec- tively, 1, 2; 1). Let us assign the cusps of X the labels a 1 , · · · , a r . For each i ∈ {1, · · · , r}, we assign to the cusp of X 2/1 that corresponds nat- urally to the cusp of X labeled a i the label b i . Thus, there is precisely one cusp of X 2/1 that has not been assigned a label ∈ {b 1 , · · · , b r }; we assign to this uniquely determined cusp the label b r+1 . For each i ∈ {1, · · · , r + 1}, we assign to the cusp of X 3/2 that corresponds nat- urally to the cusp of X 2/1 labeled b i the label c i . Thus, there is precisely one cusp of X 3/2 that has not been assigned a label ∈ {c 1 , · · · , c r+1 }; we assign to this uniquely determined cusp the label c r+2 . Now it follows from assertion (iii) in the case where n = 2, applied to the restriction of def α to Π 3/1 = Ker(q 1 ), together with [CbTpI], Theorem A, (ii), that α is FwC-admissible. Write q 12 : Π 3  Π 2 for the surjection induced by the def projection X 3 → X 2 to the factors labeled 1, 2; Π 3/2 = Ker(q 12 ) ⊆ Π 3 . Thus, to verify the fact that α is FC-admissible, it suffices to verify the following assertion: Claim 2.3.D: If I c r+2 ⊆ Π 3/2 is a cuspidal inertia sub- group associated to the cusp labeled c r+2 , then α(I c r+2 ) is a cuspidal inertia subgroup. To verify Claim 2.3.D, let us fix a cusp labeled b ∈ {b 1 , · · · , b r } [where we recall that r  = 0], a cuspidal inertia subgroup I b ⊆ Π 2/1 as- sociated to the cusp labeled b of X 2/1 , and a cuspidal inertia subgroup I c r+2 ⊆ Π 3/2 associated to the cusp labeled c r+2 of Π 3/2 . Now observe 40 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI that since the [necessarily FC-admissible] automorphism of Π 2/1 in- duced by α induces the identity automorphism on the set of conjugacy classes of cuspidal inertia subgroups of Π 2/1 , to verify the assertion that α(I c r+2 ) is a cuspidal inertia subgroup, we may assume without loss of generality [by replacing α by a suitable Π 3 -conjugate of α] that the automorphism of Π 2/1 induced by α fixes I b . Let Π E b ⊆ Π 3/2 be a minor verticial subgroup, relative to the two-dimensional configuration space X 3/1 associated to the hyperbolic curve X 2/1 , at the cusp labeled b [cf. [CmbCsp], Definition 1.4, (ii)] such that I c r+2 ⊆ Π E b . Then it fol- lows immediately from Lemma 2.2, (ii), that α fixes the Π 3/2 -conjugacy class of Π E b , i.e., that Π † E b = α(Π E b ) is a Π 3/2 -conjugate of Π E b . Thus, one verifies easily that, to verify that α(I c r+2 ) is a cuspidal inertia sub- ∼ group, it suffices to verify that the isomorphism Π E b → Π † E b induced by α is group-theoretically cuspidal — cf. [CmbGC], Definition 1.4, (iv). [Note that it follows immediately from the various definitions involved that Π E b and Π † E b may be regarded as pro-Σ fundamental groups of semi-graphs of anabelioids of pro-Σ PSC-type.] On the other hand, it follows immediately from a similar argument to the argument applied in the discussion concerning the isomorphism of the second display of [CmbCsp], Definition 1.4, (ii), that the composites def Π E b , Π † E b → Π 3/2  Π 2/1 — where the second arrow is the surjection determined by the surjec- tion q 13 : Π 3  Π 2 induced by the projection X 3 → X 2 to the factors labeled 1, 3 — are injective, and that the Π 2/1 -conjugacy class of the image in Π 2/1 of either of these composite injections coincides with the Π 2/1 -conjugacy class of a minor verticial subgroup at the cusp labeled a i [where we write b = b i — cf. [CmbCsp], Definition 1.4, (ii)]. In particular, since the automorphism of Π 2 induced by α relative to q 13 is FC-admissible [cf. [CbTpI], Theorem A, (ii)], it follows immediately ∼ that the isomorphism Π E b → Π † E b induced by α is group-theoretically cuspidal. This completes the proof of Claim 2.3.D, hence also of asser- tion (ii). Now assertion (iii) in the case where n  = 2 follows immediately from assertion (ii), together with the natural inclusions Out FC (Π n ) ⊆ Out FwC (Π n ) ⊆ Out F (Π n ) [cf. Remark 2.1.1]. This completes the proof of assertion (iii). Next, we verify assertion (i). The bijectivity portion of assertion (i) follows from assertion (ii), together with the bijectivity portion of [NodNon], Theorem B. Thus, it suffices to verify the injectivity portion of assertion (i). First, we observe that injectivity in the case where (g, r) = (0, 3) follows from assertion (ii), together with the injectiv- ity portion of [NodNon], Theorem B. Write Π † 2 (respectively, Π † 1 ) for the kernel of the surjection Π n+1  Π n−1 (respectively, Π n  Π n−1 ) COMBINATORIAL ANABELIAN TOPICS II 41 induced by the projection obtained by forgetting the factor(s) labeled def n+1, n (respectively, n). Here, if n = 1, then we set Π n−1 = Π 0 = {1}. Then recall [cf. e.g., the proof of [CmbCsp], Theorem 4.1, (i)] that we have natural isomorphisms out out Π n+1 ≃ Π † 2  Π n−1 ; Π n ≃ Π † 1  Π n−1 [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. Also, we recall [cf. [MzTa], Proposition 2.4, (i)] that one may interpret the surjection Π † 2  Π † 1 induced by the surjection Π n+1  Π n in question as the surjection “Π 2  Π 1 ” that arises from the projection X 2 → X in the case of an “X” of type (g, r +n−1). Moreover, one verifies easily that this interpretation is compatible with the definition of the various “Out(−)’s” involved. Thus, since n inj = 2 if r = 0, the above natural isomorphisms allow one to reduce the injectivity in question to the case where n = 1 and r  = 0. On the other hand, this injectivity follows immediately from a similar argument to the argument used in the proof of [CmbCsp], Corollary 2.3, (ii), by replacing [CmbCsp], Proposition 1.2, (iii) (respectively, the non-resp’d portion of [CmbCsp], Proposition 1.3, (iv); [CmbCsp], Corollary 1.12, (i)), in the proof of [CmbCsp], Corollary 2.3, (ii), by Lemma 2.2, (i) (respectively, Lemma 2.2, (ii); the injectivity in question in the case where (g, r) = (0, 3), which was verified above). This completes the proof of the injectivity portion of assertion (i), hence also of assertion (i). Finally, assertion (iv) follows immediately from assertion (i), to- gether with a similar argument to the argument applied in the proof of [CmbCsp], Theorem 4.1, (iv). This completes the proof of Theo- rem 2.3.  Corollary 2.4 (PFC-admissibility of outomorphisms). In the no- tation of Theorem 2.3, write Out PF (Π n ) ⊆ Out(Π n ) for the subgroup of PF-admissible outomorphisms [i.e., roughly speak- ing, outomorphisms that preserve the fiber subgroups up to a possible permutation of the factors — cf. [CbTpI], Definition 1.4, (i)] and Out PFC (Π n ) ⊆ Out PF (Π n ) for the subgroup of PFC-admissible outomorphisms [i.e., roughly speak- ing, outomorphisms that preserve the fiber subgroups and the cuspidal inertia subgroups up to a possible permutation of the factors — cf. [CbTpI], Definition 1.4, (iii)]. Let us regard the symmetric group on n letters S n as a subgroup of Out(Π n ) via the natural inclusion of The- orem 2.3, (iv). Finally, suppose that (g, r) ∈ {(0, 3); (1, 1)}. Then the following hold: 42 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (i) We have an equality Out(Π n ) = Out PF (Π n ). If, moreover, (r, n)  = (0, 2), then we have equalities Out(Π n ) = Out PF (Π n ) = Out F (Π n ) × S n [cf. the notational conventions introduced in Theorem 2.3]. (ii) If either r > 0, n ≥ 3 or n ≥ 4, then we have equalities Out(Π n ) = Out PFC (Π n ) = Out FC (Π n ) × S n [cf. the notational conventions introduced in Theorem 2.3]. Proof. First, we verify assertion (i). The equality in the first display of assertion (i) follows from [MzTa], Corollary 6.3, together with the assumption that (g, r) ∈ {(0, 3); (1, 1)}. The second equality in the second display of assertion (i) follows from Theorem 2.3, (iv). This completes the proof of assertion (i). Next, we verify assertion (ii). The first equality of assertion (ii) follows immediately from Theorem 2.3, (ii), together with the first equality of assertion (i). The second equality of assertion (ii) follows from [NodNon], Theorem B. This completes the proof of assertion (ii).  Corollary 2.5 (Anabelian properties of hyperbolic curves and associated configuration spaces I). Let Σ be a set of prime numbers which is either equal to the set of all prime numbers or of cardinality one; m ≤ n positive integers; (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0; k a field of characteristic ∈ Σ; k a separable closure of k; X a hyperbolic curve of type (g, r) over k. Write def G k = Gal(k/k). For each positive integer i, write X i for the i-th def configuration space of X; (X i ) k = X i × k k; Δ X i for the maximal pro-Σ quotient of the étale fundamental group of (X i ) k ; ρ Σ X i : G k −→ Out(Δ X i ) for the pro-Σ outer Galois representation associated to X i ; S i for the symmetric group on i letters; Φ i : S i −→ Out(Δ X i ) for the outer representation arising from the permutations of the factors of X i . Suppose that the following conditions are satisfied: COMBINATORIAL ANABELIAN TOPICS II 43 (1) (g, r) ∈ {(0, 3); (1, 1)}. (2) If (r, n, m) ∈ {(0, 2, 1); (0, 2, 2); (0, 3, 1)}, then there exists an l ∈ Σ such that k is l-cyclotomically full, i.e., the l-adic cyclotomic character of G k has open image. Then the following hold: (i) Let α ∈ Out(Δ X n ). Then there exists a unique element σ α ∈ S n such that α ◦ Φ n (σ α ) ∈ Out F (Δ X n ) [cf. the notational conventions introduced in Theorem 2.3]. Write α m ∈ Out F (Δ X m ) for the outomorphism of Δ X m induced by α ◦ Φ n (σ α ), relative to the quotient Δ X n  Δ X m by a fiber subgroup of co-length m of Δ X n . [Note that it follows from [CbTpI], Theorem A, (i), that α m does not depend on the choice of fiber subgroup of co-length m of Δ X n .] (ii) If (r, n, m) ∈ {(0, 2, 1); (0, 2, 2); (0, 3, 1)}, then PFC C Out(Δ Xn ) (Im(ρ Σ (Δ X n ) X n )) ⊆ Out [cf. the notational conventions introduced in Corollary 2.4]. (iii) The map Out(Δ X n ) −→ Out(Δ X m ) α → α m [cf. (i)] determines an exact sequence of homomorphisms of profinite groups Φ n 1 −→ S n −→ Out PFC (Δ X n ) −→ Out(Δ X m ) — where the second arrow is a split injection whose image commutes with Out FC (Δ X n ) and has trivial intersection with Im(ρ Σ X n ). If (r, n)  = (0, 2), then the map α → α m deter- mines a sequence of homomorphisms of profinite groups Φ n Out(Δ X n ) −→ Out(Δ X m ) 1 −→ S n −→ — where the second arrow is a split injection whose im- age commutes with Out F (Δ X n ) and has trivial intersec- tion with Im(ρ Σ X n ) — which is exact if, moreover, (r, n, m)  = (0, 3, 1). (iv) Let α ∈ Out(Δ X n ). If (r, n, m) ∈ {(0, 2, 1); (0, 3, 1)}, then we suppose further that α ∈ Out PFC (Δ X n ), which is the case if, for instance, α ∈ C Out(Δ Xn ) (Im(ρ Σ X n )) [cf. (ii)]. Then it holds that α ∈ Z Out(Δ Xn ) (Im(ρ Σ X n )) Σ (respectively, N Out(Δ Xn ) (Im(ρ Σ X n )) ; C Out(Δ Xn ) (Im(ρ X n ))) 44 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI if and only if α m ∈ Z Out(Δ Xm ) (Im(ρ Σ X m )) Σ (respectively, N Out(Δ Xm ) (Im(ρ Σ X m )) ; C Out(Δ Xm ) (Im(ρ X m ))) . (v) For each positive integer i, write Aut k (X i ) for the group of au- tomorphisms of X i over k. Then if the natural homomorphism Aut k (X m ) −→ Z Out(Δ Xm ) (Im(ρ Σ X m )) is bijective, then the natural homomorphism Aut k (X n ) −→ Z Out(Δ Xn ) (Im(ρ Σ X n )) is bijective. (vi) For each positive integer i, write Aut((X i ) k /k) for the group of automorphisms of (X i ) k that are compatible with some au- tomorphism of k; Aut ρ (G k ) for the group of automorphisms of G k that preserve Ker(ρ Σ X 1 ) ⊆ G k [where we note that, by [NodNon], Corollary 6.2, (i), for any positive integer i, it holds Σ that Ker(ρ Σ X 1 ) = Ker(ρ X i )]. Then if the natural homomorphism Aut((X m ) k /k) −→ Aut ρ (G k )× Aut(Im(ρ Σ X )) N Out(Δ Xm ) (Im(ρ Σ X m )) m is bijective, then the natural homomorphism Aut((X n ) k /k) −→ Aut ρ (G k ) × Aut(Im(ρ Σ X )) N Out(Δ Xn ) (Im(ρ Σ X n )) n is bijective. Proof. First, we verify assertion (i). The existence of such a σ α follows from the fact that Out(Δ X n ) = Out PF (Δ X n ) [cf. Corollary 2.4, (i), to- gether with assumption (1)]. The uniqueness of such a σ α follows imme- diately from the easily verified faithfulness of the action of S n , via Φ n , on the set of fiber subgroups of Δ X n . This completes the proof of asser- tion (i). Next, we verify assertion (ii). Since Out(Δ X n ) = Out PF (Δ X n ) [cf. Corollary 2.4, (i), together with assumption (1)], assertion (ii) follows immediately from [CmbGC], Corollary 2.7, (i), together with condition (2). This completes the proof of assertion (ii). Next, we verify assertion (iii). First, let us observe that it fol- lows immediately from the various definitions involved that Im(Φ n ) ⊆ Out PFC (Δ X n ). Now since Out(Δ X n ) = Out PF (Δ X n ) [cf. Corollary 2.4, (i), together with assumption (1)], and Out F (Δ X n ) is normalized by Out PF (Δ X n ), one verifies easily [i.e., by considering the action of ele- ments of Out PF (Δ X n ) on the set of fiber subgroups of Δ X n ] that the second arrow in either of the two displayed sequences is a split injec- tion. Moreover, since [as is easily verified] the outer action of G k , via ρ Σ X n , on Δ X n fixes every fiber subgroup of Δ X n , it follows immediately from the faithfulness of the action of S n , via Φ n , on the set of fiber subgroups of Δ X n that the image of the second arrow in either of the COMBINATORIAL ANABELIAN TOPICS II 45 two displayed sequences has trivial intersection with Im(ρ Σ X n ). Now it follows from [NodNon], Theorem B, that the image of the second arrow of the first displayed sequence commutes with Out FC (Δ X n ); in partic- ular, one verifies easily from the various definitions involved [cf. also Corollary 2.4, (i), together with assumption (1)] that the third arrow of the first displayed sequence is a homomorphism. If (r, n)  = (0, 2), then it follows from Corollary 2.4, (i), together with assumption (1), that the image of the second arrow of the second displayed sequence commutes with Out F (Δ X n ); in particular, one verifies easily from the various def- initions involved [cf. also Corollary 2.4, (i), together with assumption (1)] that the third arrow of the second displayed sequence is a homo- morphism. Now if (r, m)  = (0, 1), then it follows immediately from the injectivity portion of Theorem 2.3, (i), together with the equality Out(Δ X n ) = Out PF (Δ X n ) [cf. Corollary 2.4, (i), together with assump- tion (1)], that the kernel of the third arrow in either of the two displayed sequences is Im(Φ n ). Moreover, if (r, n, m) ∈ {(0, 2, 1); (0, 3, 1)}, then it follows immediately from the injectivity portion of [NodNon], Theo- rem B, that the kernel of the third arrow in the first displayed sequence is Im(Φ n ). On the other hand, if (r, m) = (0, 1) and n ∈ {2, 3}, then it follows immediately from the injectivity portion of [NodNon], Theorem B, together with Corollary 2.4, (ii), together with assumption (1), that the kernel of the third arrow in either of the two displayed sequences is Im(Φ n ). This completes the proof of assertion (iii). Next, we verify assertion (iv). Now since the permutations of the factors of X n give rise to automorphisms of X n over k, it follows im- mediately that Im(Φ n ) ⊆ Z Out(Δ Xn ) (Im(ρ Σ X n )). In particular, to verify assertion (iv), we may assume without loss of generality — by replacing α by α n [cf. assertion (i)] — that α ∈ Out F (Δ X n ), and that m < n. Then necessity follows immediately. On the other hand, sufficiency follows immediately from the exact sequences of assertion (iii). This completes the proof of assertion (iv). Assertion (v) (respectively, (vi)) follows immediately from assertions (i), (ii), (iii), (iv), together with Lemma 2.7, (iii), below (respectively, Lemma 2.7, (iv), below). This completes the proof of Corollary 2.5.  Corollary 2.6 (Anabelian properties of hyperbolic curves and associated configuration spaces II). Let Σ be a set of prime num- bers which is either equal to the set of all prime numbers or of cardi- nality one; m ≤ n positive integers; (g X , r X ), (g Y , r Y ) pairs of non- negative integers such that 2g X − 2 + r X , 2g Y − 2 + r Y > 0; k X , k Y fields; k X , k Y separable closures of k X , k Y , respectively; X, Y hy- perbolic curves of type (g X , r X ), (g Y , r Y ) over k X , k Y , respectively. def def Write G k X = Gal(k X /k X ); G k Y = Gal(k Y /k Y ). For each positive integer i, write X i , Y i for the i-th configuration spaces of X, Y , 46 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI def def respectively; (X i ) k X = X i × k X k X ; (Y i ) k Y = Y i × k Y k Y ; π 1 Σ ((X i ) k X ), π 1 Σ ((Y i ) k Y ) for the maximal pro-Σ quotients of the étale fundamental (Σ) groups π 1 ((X i ) k X ), π 1 ((Y i ) k Y ) of (X i ) k X , (Y i ) k Y , respectively; π 1 (X i ), (Σ) π 1 (Y i ) for the geometrically pro-Σ étale fundamental groups of X i , Y i , respectively, i.e., the quotients of the étale fundamental groups π 1 (X i ), π 1 (Y i ) of X i , Y i by the respective kernels of the natural surjections π 1 ((X i ) k X )  π 1 Σ ((X i ) k X ), π 1 ((Y i ) k Y )  π 1 Σ ((Y i ) k Y ). Suppose that the following conditions are satisfied: (1) {(g X , r X ); (g Y , r Y )} ∩ {(0, 3); (1, 1)} = ∅. (2) If (r X , n, m) (respectively, (r Y , n, m)) is contained in the set {(0, 2, 1); (0, 2, 2); (0, 3, 1)}, then there exists an l ∈ Σ such that k X (respectively, k Y ) is l-cyclotomically full, i.e., the l- adic cyclotomic character of G k X (respectively, G k Y ) has open image. Then the following hold: ∼ (i) Let θ : k X → k Y be an isomorphism of fields that determines ∼ an isomorphism k X → k Y . For each positive integer i, write Isom θ (X i , Y i ) for the set of isomorphisms of X i with Y i that ∼ are compatible with the isomorphism k X → k Y determined (Σ) (Σ) by θ; Isom θ (π 1 (X i ), π 1 (Y i )) for the set of isomorphisms of (Σ) (Σ) π 1 (X i ) with π 1 (Y i ) that are compatible with the isomorphism ∼ G k X → G k Y determined by θ. Then if the natural map (Σ) (Σ) Isom θ (X m , Y m ) −→ Isom θ (π 1 (X m ), π 1 (Y m ))/Inn(π 1 Σ ((Y m ) k Y )) is bijective, then the natural map (Σ) (Σ) Isom θ (X n , Y n ) −→ Isom θ (π 1 (X n ), π 1 (Y n ))/Inn(π 1 Σ ((Y n ) k Y )) is bijective. (ii) For each positive integer i, write Isom((X i ) k X /k X , (Y i ) k Y /k Y ) for the set of isomorphisms of (X i ) k X with (Y i ) k Y that are com- patible with some field isomorphism of k X with k Y ; (Σ) (Σ) Isom(π 1 (X i )/G k X , π 1 (Y i )/G k Y ) (Σ) (Σ) for the set of isomorphisms of π 1 (X i ) with π 1 (Y i ) that are compatible with some isomorphism of G k X with G k Y . Then if the natural map Isom((X m ) k X /k X , (Y m ) k Y /k Y ) (Σ) (Σ) −→ Isom(π 1 (X m )/G k X , π 1 (Y m )/G k Y )/Inn(π 1 Σ ((Y m ) k Y )) is bijective, then the natural map Isom((X n ) k X /k X , (Y n ) k Y /k Y ) COMBINATORIAL ANABELIAN TOPICS II (Σ) 47 (Σ) −→ Isom(π 1 (X n )/G k X , π 1 (Y n )/G k Y )/Inn(π 1 Σ ((Y n ) k Y )) is bijective. Proof. Consider assertion (i) (respectively, (ii)). If the set (Σ) (Σ) Isom θ (π 1 (X n ), π 1 (Y n ))/Inn(π 1 Σ ((Y n ) k Y )) (respectively, (Σ) (Σ) Isom(π 1 (X n )/G k X , π 1 (Y n )/G k Y )/Inn(π 1 Σ ((Y n ) k Y )) ) is empty, then assertion (i) (respectively, (ii)) is immediate. Thus, we may suppose without loss of generality that this set is nonempty. Then one verifies easily from [MzTa], Corollary 6.3, together with condition (1), that the set (Σ) (Σ) Isom θ (π 1 (X m ), π 1 (Y m ))/Inn(π 1 Σ ((Y m ) k Y )) (respectively, (Σ) (Σ) Isom(π 1 (X m )/G k X , π 1 (Y m )/G k Y )/Inn(π 1 Σ ((Y m ) k Y )) ) is nonempty. Thus, it follows immediately from the bijectivity assumed in assertion (i) (respectively, (ii)) that there exists an isomorphism ∼ ∼ X m → Y m that is compatible with the isomorphism k X → k Y deter- ∼ mined by θ (respectively, an isomorphism (X m ) k X → (Y m ) k Y that is ∼ compatible with some isomorphism k X → k Y ). In particular, it follows immediately from Lemma 2.7, (iii), below (respectively, Lemma 2.7, ∼ (iv), below) that there exists an isomorphism X → Y that is compatible ∼ with the isomorphism k X → k Y determined by θ (respectively, an iso- ∼ morphism X × k X k X → Y × k Y k Y that is compatible with some isomor- ∼ phism k X → k Y ). Thus, by pulling back the various objects involved via this isomorphism, to verify assertion (i) (respectively, (ii)), we may assume without loss of generality that (X, k X , k X , θ) = (Y, k Y , k Y , id k X ) (respectively, (X, k X , k X ) = (Y, k Y , k Y )). Then assertion (i) (respec- tively, (ii)) follows from Corollary 2.5, (v) (respectively, Corollary 2.5, (vi)). This completes the proof of Corollary 2.6.  Lemma 2.7 (Isomorphisms between configuration spaces of hyperbolic curves). Let n be a positive integer; (g X , r X ), (g Y , r Y ) pairs of nonnegative integers such that 2g X − 2 + r X , 2g Y − 2 + r Y > 0; k X , k Y fields; k X , k Y separable closures of k X , k Y , respectively; X, Y hyperbolic curves of type (g X , r X ), (g Y , r Y ) over k X , k Y , respectively. Write X n , Y n for the n-th configuration spaces of X, Y , respectively; def def def def X k X = X × k X k X ; Y k Y = Y × k Y k Y ; (X n ) k X = X n × k X k X ; (Y n ) k Y = 48 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Y n × k Y k Y ; S n for the symmetric group on n letters; Aut k X (X n ) for the group of automorphisms of X n over k X ; Ψ n : S n −→ Aut k X (X n ) for the action of S n on X n over k X obtained by permuting the factors of X n . Suppose that (g X , r X ), (g Y , r Y ) ∈ {(0, 3); (1, 1)}. Then the following hold: ∼ (i) Let α : X n → Y n be an isomorphism. Then there exists a ∼ unique isomorphism α 0 : k Y → k X that is compatible with α relative to the structure morphisms of X n , Y n . ∼ (ii) Let α : X n → Y n be an isomorphism. Then there exist a unique permutation σ ∈ Ψ n (S n ) ⊆ Aut k X (X n ) and a unique isomor- ∼ phism α 1 : X → Y that is compatible with α ◦ σ relative to the projections X n → X, Y n → Y to each of the n factors. (iii) Write Isom(X n , Y n ) for the set of isomorphisms of X n with Y n ; def Isom(X, Y ) = Isom(X 1 , Y 1 ). Then the natural map Isom(X, Y ) × Ψ n (S n ) −→ Isom(X n , Y n ) is bijective. (iv) Write Isom((X n ) k X /k X , (Y n ) k Y /k Y ) for the set of isomorphisms ∼ (X n ) k X → (Y n ) k Y that are compatible with some isomorphism ∼ def k Y → k X ; Isom(X k X /k X , Y k Y /k Y ) = Isom((X 1 ) k X /k X , (Y 1 ) k Y /k Y ). Then the natural map Isom(X k X /k X , Y k Y /k Y ) × Ψ n (S n ) −→ Isom((X n ) k X /k X , (Y n ) k Y /k Y ) is bijective. Proof. First, we verify assertion (i). Write (C n X ) log , (C n Y ) log for the n-th log configuration spaces [cf. the discussion entitled “Curves” in “Notations and Conventions”] of [the smooth log curves over k X , k Y determined by] X, Y , respectively. Then recall [cf. the discussion at the beginning of [MzTa], §2] that (C n X ) log , (C n Y ) log are log regular log schemes whose interiors are naturally isomorphic to X n , Y n , respec- tively, and that the underlying schemes C n X , C n Y of (C n X ) log , (C n Y ) log are proper over k X , k Y , respectively. Thus, by applying [ExtFam], Theorem A, (1), to the composite α X n → Y n → C n Y → M g Y ,r Y +n — where we refer to the discussion entitled “Curves” in [CbTpI], §0, concerning the notation “M g Y ,r Y +n ”; the third arrow is the natural (1-)morphism arising from the definition of C n Y — we conclude that the composite α X n → Y n → C n Y → M g Y ,r Y +n → (M g Y ,r Y +n ) c COMBINATORIAL ANABELIAN TOPICS II 49 — where we write (M g Y ,r Y +n ) c for the coarse moduli space associated to M g Y ,r Y +n — factors through the natural open immersion X n → C n X . On the other hand, one verifies immediately that the composite C n Y → M g Y ,r Y +n → (M g Y ,r Y +n ) c is proper and quasi-finite, hence finite. In particular, if we write C Γ ⊆ C n X × k C n Y for the scheme-theoretic clo- α sure of the graph of the composite X n → Y n → C n Y , then the composite pr C Γ → C n X × k C n Y → 1 C n X is a finite morphism from an integral scheme to a normal scheme which induces an isomorphism between the respective function fields. Thus, we conclude that this composite is an isomor- phism, hence that α extends uniquely to a morphism C n X → C n Y . Now recall that C n X is proper, geometrically normal, and geometrically con- nected over k X . Thus, one verifies immediately, by considering global sections of the respective structure sheaves, that there exists a unique homomorphism α 0 : k Y → k X that is compatible with α. Moreover, by applying a similar argument to α −1 , it follows that α 0 is an isomor- phism. This completes the proof of assertion (i). Next, we verify assertion (ii). First, let us observe that, by replacing ∼ Y by the result of base-changing Y via α 0 : k Y → k X [cf. assertion (i)], we may assume without loss of generality that k Y = k X , k Y = k X , and that α is an isomorphism over k X . Next, let us observe that it is immediate that σ and α 1 as in the statement of assertion (ii) are unique; thus, it remains to verify the existence of such σ and α 1 . Next, let us observe that it follows immediately from [MzTa], Corollary 6.3, that there exists a permutation σ ∈ Ψ n (S n ) such that if we identify the respective sets of fiber subgroups of Δ X n , Δ Y n — where we write Δ X n , Δ Y n for the maximal pro-l quotients of the étale fundamental groups of (X n ) k X , (Y n ) k X , respectively, for some prime number l that is invertible in k X — with the set 2 {1,··· ,n} [cf. the discussion entitled “Sets” in [CbTpI], §0] in the evident way, then the automorphism of def the set 2 {1,··· ,n} induced by the composite β = α ◦ σ is the identity automorphism. Write pr X : X n → X, pr Y : Y n → Y for the projections to the factor labeled n, respectively. Then we claim that the following assertion holds: ∼ Claim 2.7.A: There exists an isomorphism α 1 : X → Y that is compatible with β relative to pr X , pr Y . Indeed, write Γ ⊆ X × k X Y for the scheme-theoretic image via X n × k X (pr ,id Y ) β pr X Y −→ X × k X Y of the graph of the composite X n → Y n → Y Y . Next, let us observe that if Z is an irreducible scheme of finite type over k X , then any nonconstant [i.e., dominant] k X -morphism Z → Y k X induces an open homomorphism between the respective fundamental groups. Thus, since the automorphism of the set 2 {1,··· ,n} induced by β is the identity automorphism, it follows immediately that, for any k X -valued geometric point x of X, if we write F for the geometric 50 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI β k fiber of pr X : X n → X at x, then the composite F → (X n ) k X → X (pr Y ) k (Y n ) k X → X Y k X is constant. In particular, one verifies immediately that Γ is an integral, separated scheme of dimension 1. Thus, since pr X is surjective, geometrically connected, smooth, and factors through pr the composite Γ → X × k X Y → 1 X, it follows immediately that this composite morphism Γ → X is surjective and induces an isomorphism between the respective function fields. Therefore, one concludes easily, by applying Zariski’s main theorem, that the composite Γ → X × k X pr Y → 1 X is an isomorphism, hence that there exists a unique morphism α 1 : X → Y such that pr Y ◦ β = α 1 ◦ pr X . Moreover, by applying a similar argument to β −1 , it follows that α 1 is an isomorphism. This completes the proof of Claim 2.7.A. ∼ Write γ for the composite of β with the isomorphism Y n → X n de- termined by α 1 −1 . Then it is immediate that γ is an automorphism of X n over X relative to pr X ; in particular, the outomorphism of Δ X n induced by γ is contained in the kernel of the homomorphism Out F (Δ X n ) → Out F (Δ X ) — where we write Δ X for the maximal pro-l quotient of the étale fundamental group of X k X — induced by pr X . Now, by applying a similar argument to the argument of the proof of Claim 2.7.A, one verifies easily that, for each i ∈ {1, · · · , n}, there ex- ists an automorphism γ 1,i of X that is compatible with γ relative to the projection X n → X to the factor labeled i. [Thus, γ 1,n = id X .] More- over, since, by applying induction on n, we may assume that assertion (ii) has already been verified for n − 1, it follows immediately that the outomorphism of Δ X n induced by γ is contained in Out FC (Δ X n ), hence in the kernel of the homomorphism Out FC (Δ X n ) → Out FC (Δ X ) induced by the projections X n → X to each of the n factors [cf. [CmbCsp], Proposition 1.2, (iii)]. Therefore, it follows immediately from the argument of the first paragraph of the proof of [LocAn], The- orem 14.1, that, for each i ∈ {1, · · · , n}, γ 1,i is the identity automor- phism of X, hence also that γ is the identity automorphism of X n . This completes the proof of assertion (ii). Assertions (iii), (iv) follow immediately from assertion (ii), together with the various definitions involved. This completes the proof of Lemma 2.7.  COMBINATORIAL ANABELIAN TOPICS II 51 3. Synchronization of tripods In the present §3, we introduce and study the notion of a tripod of the log fundamental group of the log configuration space of a stable log curve [cf. Definition 3.3, (i), below]. In particular, we discuss the phenomenon of synchronization among the various tripods of the log fundamental group [cf. Theorems 3.17; 3.18, below]. One interesting consequence of this phenomenon of tripod synchronization is a certain non-surjectivity result [cf. Corollary 3.22 below]. Finally, we apply the theory of synchronization of tripods to show that, under certain conditions, commuting profinite Dehn multi-twists are “co-Dehn” [cf. Corollary 3.25 below] and to compute the commensurator of certain purely combinatorial/group-theoretic groups of profinite Dehn multi- twists in terms of scheme theory [cf. Corollary 3.27 below]. In the present §3, let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; n a positive integer; Σ a set of prime numbers which is either the set of all prime numbers or of cardinality one; k an algebraically closed field of characteristic ∈ Σ; (Spec k) log the log scheme obtained by equipping Spec k with the log structure determined by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over (Spec k) log . For each [possibly empty] subset E ⊆ {1, · · · , n}, write X E log for the #E-th log configuration space of the stable log curve X log [cf. the discussion entitled “Curves” in “Notations and Conventions”], where we think of the factors as being labeled by the elements of E ⊆ {1, · · · , n}; Π E for the maximal pro-Σ quotient of the kernel of the natural surjection π 1 (X E log )  π 1 ((Spec k) log ). Thus, by applying a suitable specializa- tion isomorphism — cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1 — one verifies easily that Π E is equipped with a natural structure of pro-Σ configuration space group — cf. [MzTa], Definition 2.3, (i). For each 1 ≤ m ≤ n, write def def log log = X {1,··· X m ,m} ; Π m = Π {1,··· ,m} . Thus, for subsets E  ⊆ E ⊆ {1, · · · , n}, we have a projection log log p log E/E  : X E → X E  obtained by forgetting the factors that belong to E \ E  . For E  ⊆ E ⊆ {1, · · · , n} and 1 ≤ m  ≤ m ≤ n, we shall write p Π E/E  : Π E  Π E  52 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI for some fixed surjection [that belongs to the collection of surjections that constitutes the outer surjection] induced by p log E/E  ; def Π E/E  = Ker(p Π E/E  ) ⊆ Π E ; def log log log p log m/m  = p {1,··· ,m}/{1,··· ,m  } : X m −→ X m  ; def Π p Π m/m  = p {1,··· ,m}/{1,··· ,m  } : Π m  Π m  ; def Π m/m  = Π {1,··· ,m}/{1,··· ,m  } ⊆ Π m . Finally, recall [cf. the statement of Theorem 2.3, (iv)] the natural inclusion S n → Out(Π n ) — where we write S n for the symmetric group on n letters — obtained by permuting the various factors of X n . We shall regard S n as a subgroup of Out(Π n ) by means of this natural inclusion. Definition 3.1. Let i ∈ E ⊆ {1, · · · , n}; x ∈ X n (k) a k-valued geo- metric point of the underlying scheme X n of X n log . (i) Let E  ⊆ {1, · · · , n} be a subset. Then we shall write x E  ∈ X E  (k) for the k-valued geometric point of X E  obtained by forming the image of x ∈ X n (k) via p {1,··· ,n}/E  : X n → X E  ; def log x log E  = x E  × X E  X E  . (ii) We shall write G for the semi-graph of anabelioids of pro-Σ PSC-type deter- mined by the stable log curve X log over (Spec k) log [cf. [CmbGC], Example 2.5]; G for the underlying semi-graph of G; Π G for the [pro-Σ] fundamental group of G; G  −→ G for the universal covering of G corresponding to Π G . Thus, we have a natural outer isomorphism ∼ Π 1 −→ Π G . Throughout our discussion of the objects introduced at the ∼ beginning of the present §3, let us fix an isomorphism Π 1 → Π G that belongs to the collection of isomorphisms that constitutes the above natural outer isomorphism. COMBINATORIAL ANABELIAN TOPICS II 53 (iii) We shall write G i∈E,x for the semi-graph of anabelioids of pro-Σ PSC-type deter- log mined by the geometric fiber of the projection p log E/(E\{i}) : X E → log log over x log X E\{i} E\{i} → X E\{i} [cf. (i)]; Π G i∈E,x for the [pro-Σ] fundamental group of G i∈E,x . Thus, we have a natural identification G = G i∈{i},x and a natural Π E -orbit [i.e., relative to composition with au- tomorphisms induced by conjugation by elements of Π E ] of isomorphisms ∼ (Π E ⊇) Π E/(E\{i}) −→ Π G i∈E,x . Throughout our discussion of the objects introduced at the beginning of the present §3, let us fix an outer isomorphism ∼ Π E/(E\{i}) −→ Π G i∈E,x whose constituent isomorphisms belong to the Π E -orbit of iso- morphisms just discussed. (iv) Let v ∈ Vert(G i∈E,x ) (respectively, e ∈ Cusp(G i∈E,x ); e ∈ Node(G i∈E,x ); e ∈ Edge(G i∈E,x ); z ∈ VCN(G i∈E,x )). Then we shall refer to the image [in Π E ] of a verticial (respectively, a cuspidal; a nodal; an edge-like; a VCN-) subgroup [cf. [CbTpI], Definition 2.1, (i)] of Π G i∈E,x associated to v (respectively, e; e; ∼ e; z) via the inverse Π G i∈E,x → Π E/(E\{i}) ⊆ Π E of any isomor- phism that lifts the fixed outer isomorphism discussed in (iii) as a verticial (respectively, a cuspidal; a nodal; an edge-like; a VCN-) subgroup of Π E associated to v (respectively, e; e; e; z). Thus, the notion of a verticial (respectively, a cuspidal; a nodal; an edge-like; a VCN-) subgroup of Π E associated to v (respectively, e; e; e; z) depends on the choice of the fixed outer isomorphism of (iii) [but cf. Lemma 3.2, (i), below, in the case of cusps!]. (v) We shall say that a vertex v ∈ Vert(G i∈E,x ) of G i∈E,x is a(n) [E-]tripod of X n log if v is of type (0, 3) [cf. [CbTpI], Definition 2.3, (iii)]. If, in this situation, C(v)  = ∅, then we shall say that the tripod v is cusp-supporting. (vi) We shall say that a cusp c ∈ Cusp(G i∈E,x ) of G i∈E,x is diagonal if c does not arise from a cusp of the copy of X log given by the factor of X E log labeled i ∈ E. 54 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Lemma 3.2 (Cusps of various fibers). Let i ∈ E ⊆ {1, · · · , n}; x ∈ X n (k). Then the following hold: ∼ (i) Let c ∈ Cusp(G i∈E,x ) and Π c ⊆ Π G i∈E,x ← Π E/(E\{i}) a cuspidal ∼ subgroup of Π G i∈E,x ← Π E/(E\{i}) associated to c ∈ Cusp(G i∈E,x ). Then any Π E -conjugate of Π c is, in fact, a Π E/(E\{i}) -conju- gate of Π c . (ii) Each diagonal cusp of G i∈E,x [cf. Definition 3.1, (vi)] admits a natural label ∈ E \ {i}. More precisely, for each j ∈ E \ {i}, there exists a unique diagonal cusp of G i∈E,x that arises from the divisor of the fiber product over k of #E copies of X consisting of the points whose i-th and j-th factors coincide. (iii) Let α ∈ Aut F (Π n ) [cf. [CmbCsp], Definition 1.1, (ii)]. Sup- pose that either E  = {1, · · · , n} or n ≥ n FC [cf. Theorem 2.3, ∼ (ii)]. Then the outomorphism of Π G i∈E,x ← Π E/(E\{i}) deter- mined by α is group-theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)]. (iv) Let α ∈ Aut F (Π n ) and c ∈ Cusp(G i∈E,x ) a diagonal cusp of ∼ G i∈E,x . Suppose that the outomorphism of Π G i∈E,x ← Π E/(E\{i}) determined by α is group-theoretically cuspidal. Then this outomorphism preserves the Π G i∈E,x -conjugacy class of ∼ cuspidal subgroups of Π G i∈E,x ← Π E/(E\{i}) associated to c ∈ Cusp(G i∈E,x ). Proof. Assertion (i) follows immediately from the [easily verified] fact that the restriction of p Π E/(E\{i}) : Π E  Π E\{i} to the normalizer of Π c in Π E is surjective. Assertion (ii) follows immediately from the various definitions involved. Next, we verify assertion (iii). If E  = {1, · · · , n} (respectively, n ≥ n FC ), then assertion (iii) follows immediately from [CbTpI], Theorem A, (ii) (respectively, Theorem 2.3, (ii), of the present monograph), together with assertion (i). This completes the proof of assertion (iii). Finally, assertion (iv) follows immediately from the definition of F-admissibility [cf. also assertion (ii)]. This completes the proof of Lemma 3.2.  Definition 3.3. Let E ⊆ {1, · · · , n}. (i) We shall say that a closed subgroup H ⊆ Π E of Π E is a(n) [E-]tripod of Π n if H is a verticial subgroup of Π E [cf. Def- inition 3.1, (iv)] associated to a(n) [E-]tripod v of X n log [cf. Definition 3.1, (v)]. If, in this situation, the tripod v is cusp- supporting [cf. Definition 3.1, (v)], then we shall say that the tripod H is cusp-supporting. COMBINATORIAL ANABELIAN TOPICS II 55 (ii) We shall say that an E-tripod of Π n [cf. (i)] is trigonal if, for every j ∈ E, the image of the tripod via p Π E/{j} : Π E  Π {j} is trivial. (iii) Let T ⊆ Π E be an E-tripod of Π n [cf. (i)] and E  ⊆ E. Then  we shall say that T is E  -strict if the image p Π E/E  (T ) ⊆ Π E of Π  T via p E/E  : Π E  Π E  is an E -tripod of Π n , and, moreover, for every E   E  , the image of the E  -tripod p Π E/E  (T ) via Π p E  /E  : Π E   Π E  is not a tripod of Π n . (iv) Let h be a positive integer. Then we shall say that an E- tripod T of Π n [cf. (i)] is h-descendable if there exists a subset   E  ⊆ E such that the image of T via p Π E/E  : Π E  Π E is an E - tripod of Π n , and, moreover, #E  ≤ n − h. [Thus, one verifies immediately that an E-tripod T ⊆ Π E of Π n is 1-descendable if and only if either E  = {1, · · · , n} or T fails to be E-strict — cf. (iii).] Remark 3.3.1. In the notation of Definition 3.1, let v ∈ Vert(G i∈E,x ) be an E-tripod of X n log [cf. Definition 3.1, (v)] and T ⊆ Π E an E-tripod of Π n associated to v [cf. Definition 3.3, (i)]. Write F v for the irreducible component of the geometric fiber of p E/(E\{i}) : X E → X E\{i} at x E\{i} corresponding to v; F v log for the log scheme obtained by equipping F v with the log structure induced by the log structure of X E log ; n v for the rank of the group-characteristic of F v log [cf. [MzTa], Definition 5.1, (i)] at the generic point of F v . Then it is immediate that the n v -interior U v ⊆ F v of F v log [cf. [MzTa], Definition 5.1, (i)] is a nonempty open subset of F v which is isomorphic to P 1 k \ {0, 1, ∞} over k. Moreover, one verifies easily that if we write U v log for the log scheme obtained by equipping U v with the log structure induced by the log structure of X E log , then the natural morphism U v log → U v [obtained by forgetting the log ∼ structure of U v log ] determines a natural outer isomorphism T → π 1 Σ (U v ) — where we write “π 1 Σ (−)” for the maximal pro-Σ quotient of the étale fundamental group of “(−)”. In particular, we obtain a natural outer isomorphism ∼ T −→ π 1 Σ (P 1 k \ {0, 1, ∞}) that is well-defined up to composition with an outomorphism of π 1 Σ (P 1 k \ {0, 1, ∞}) that arises from an automorphism of P 1 k \ {0, 1, ∞} over k. Definition 3.4. Let E ⊆ {1, · · · , n}. (i) Let T ⊆ Π E be an E-tripod of Π n [cf. Definition 3.3, (i)]. Then T may be regarded as the “Π 1 ” that occurs in the case 56 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI where we take “X log ” to be the smooth log curve associated to P 1 k \ {0, 1, ∞} [cf. Remark 3.3.1]. We shall write Out C (T ) ⊆ Out(T ) for the [closed] subgroup of Out(T ) consisting of C-admissible outomorphisms of T [cf. [CmbCsp], Definition 1.1, (ii)]; Out C (T ) cusp ⊆ Out C (T ) for the [closed] subgroup of Out(T ) consisting of C-admissible outomorphisms of T that induce the identity automorphism of the set of T -conjugacy classes of cuspidal inertia subgroups; Out(T ) Δ ⊆ Out(T ) for the centralizer of the subgroup [≃ S 3 , where we write S 3 for the symmetric group on 3 letters] of Out(T ) consisting of the outer modular symmetries [cf. [CmbCsp], Definition 1.1, (vi)]; Out(T ) + ⊆ Out(T ) for the [closed] subgroup of Out(T ) given by the image of the natural homomorphism Out F (T 2 ) = Out FC (T 2 ) → Out(T ) [cf. Theorem 2.3, (ii); [CmbCsp], Proposition 1.2, (iii)] — where we write T 2 for the “Π 2 ” that occurs in the case where we take “X log ” to be the smooth log curve associated to P 1 k \ {0, 1, ∞}; def Out C (T ) Δ = Out C (T ) ∩ Out(T ) Δ ; def Out C (T ) Δ+ = Out C (T ) Δ ∩ Out(T ) + [cf. [CmbCsp], Definition 1.11, (i)]. (ii) Let E  ⊆ {1, · · · , n}; let T ⊆ Π E , T  ⊆ Π E  be E-, E  -tripods of Π n [cf. Definition 3.3, (i)], respectively. Then we shall say that ∼ an outer isomorphism α : T → T  is geometric if the composite ∼ α ∼ ∼ π 1 Σ (P 1 k \ {0, 1, ∞}) ←− T −→ T  −→ π 1 Σ (P 1 k \ {0, 1, ∞}) — where the first and third arrows are natural outer isomor- phisms of the sort discussed in Remark 3.3.1 — arises from an automorphism of P 1 k \ {0, 1, ∞} over k. Remark 3.4.1. In the notation of Definition 3.4, (ii), one verifies easily ∼ that every geometric outer isomorphism α : T → T  preserves cuspidal inertia subgroups and outer modular symmetries [cf. [CmbCsp], Defi- ∼ nition 1.1, (vi)], and, moreover, lifts to an outer isomorphism T 2 → T 2  [i.e., of the corresponding “Π 2 ’s”] that arises from an isomorphism of COMBINATORIAL ANABELIAN TOPICS II 57 two-dimensional configuration spaces. In particular, the isomorphism ∼ Out(T ) → Out(T  ) induced by α determines isomorphisms ∼ ∼ Out C (T ) −→ Out C (T  ) , Out C (T ) cusp −→ Out C (T  ) cusp , ∼ ∼ Out(T ) Δ −→ Out(T  ) Δ , Out(T ) + −→ Out(T  ) + [cf. Definition 3.4, (i)]. Lemma 3.5 (Triviality of the action on the set of cusps). In the notation of Definition 3.4, it holds that Out C (T ) Δ ⊆ Out C (T ) cusp . Proof. This follows immediately from the [easily verified] fact that S 3 is center-free, together with the various definitions involved.  Lemma 3.6 (Vertices, cusps, and nodes of various fibers). Let i, j ∈ E be two distinct elements of a subset E ⊆ {1, · · · , n}; x ∈ X n (k). Write z i,j,x ∈ VCN(G j∈E\{i},x ) for the element of VCN(G j∈E\{i},x ) on which x E\{i} lies, that is to say: If x E\{i} is a cusp or node of the geo- log log metric fiber of the projection p log (E\{i})/(E\{i,j}) : X E\{i} → X E\{i,j} over def x log E\{i,j} corresponding to an edge e ∈ Edge(G j∈E\{i},x ), then z i,j,x = e; if x E\{i} is neither a cusp nor a node of the geometric fiber of the pro- log log log jection p log (E\{i})/(E\{i,j}) : X E\{i} → X E\{i,j} over x E\{i,j} , but lies on the irreducible component of the geometric fiber corresponding to a vertex def v ∈ Vert(G j∈E\{i},x ), then z i,j,x = v. Then the following hold: (i) The automorphism of X E log determined by permuting the factors labeled i, j induces natural bijections ∼ Vert(G j∈E\{i},x ) −→ Vert(G i∈E\{j},x ) ; ∼ Cusp(G j∈E\{i},x ) −→ Cusp(G i∈E\{j},x ) ; ∼ Node(G j∈E\{i},x ) −→ Node(G i∈E\{j},x ) . (ii) Let us write c diag i,j,x ∈ Cusp(G i∈E,x ) for the diagonal cusp of G i∈E,x [cf. Definition 3.1, (vi)] la- log beled j ∈ E\{i} [cf. Lemma 3.2, (ii)]. Then p log E/(E\{j}) : X E → log X E\{j} induces a bijection ∼ Cusp(G i∈E,x ) \ {c diag i,j,x } −→ Cusp(G i∈E\{j},x ) . 58 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI log (iii) Suppose that z i,j,x ∈ Vert(G j∈E\{i},x ). Then p log E/(E\{j}) : X E → log X E\{j} induces a bijection ∼ Vert(G i∈E,x ) → Vert(G i∈E\{j},x ) . (iv) Suppose that z i,j,x ∈ Edge(G j∈E\{i},x ). Then there exists a unique vertex new v i,j,x ∈ Vert(G i∈E,x ) log log such that p log E/(E\{j}) : X E → X E\{j} induces a bijection ∼ new Vert(G i∈E,x ) \ {v i,j,x } → Vert(G i∈E\{j},x ) . new new is of type (0, 3) [i.e., v i,j,x is an E-tripod Moreover, v i,j,x diag log new of X n — cf. Definition 3.1, (v)], and c i,j,x ∈ C(v i,j,x ) [cf. new (ii)]. Finally, any verticial subgroup of Π E associated to v i,j,x surjects, via p Π E/(E\{j}) , onto an edge-like subgroup of Π E\{j} associated to the edge ∈ Edge(G i∈E\{j},x ) determined by z i,j,x ∈ Edge(G j∈E\{i},x ) via the bijections of (i). (v) Suppose that #E = 3. Write h ∈ E \ {i, j} for the unique element of E \ {i, j}. Suppose, moreover, that z i,j,x = c diag j,h,x ∈ Cusp(G j∈E\{i},x ) [cf. (ii)]. Then the Π E -conjugacy class of new ∈ a verticial subgroup of Π E associated to the vertex v i,j,x Vert(G i∈E,x ) [cf. (iv)] depends only on i and not on the choice of the pair (j, x). Moreover, these three Π E -conjugacy classes [cf. the dependence on the choice of i ∈ E] may also be characterized uniquely as the Π E -conjugacy classes of sub- groups of Π E associated to some trigonal E-tripod of Π n [cf. Definition 3.3, (ii)]. Proof. First, we verify assertions (i), (ii), (iii), and (iv). To verify as- sertions (i), (ii), (iii), and (iv) — by replacing X E log by the base-change log log of p log → X E\{i,j} via a suitable morphism of log schemes E\{i,j} : X E log (Spec k) log → X E\{i,j} whose image lies on x E\{i,j} ∈ X E\{i,j} (k) [cf. Definition 3.1, (i)] — we may assume without loss of generality that #E = 2. Then one verifies easily from the various definitions involved that assertions (i), (ii), (iii), and (iv) hold. This completes the proof of assertions (i), (ii), (iii), and (iv). Finally, we consider assertion (v). First, we observe the easily verified fact [cf. assertions (iii), (iv)] that the irreducible component corresponding to an E-tripod of X n log [cf. Definition 3.1, (v)] that gives rise to a trigonal E-tripod of Π n neces- sarily collapses to a point upon projection to X E  for any E  ⊆ E of cardinality ≤ 2. In light of this observation, it follows immediately [cf. assertions (i), (ii), (iii), (iv)] that any E-tripod of X n log that gives rise to new ” as described in the a trigonal E-tripod of Π n arises as a vertex “v i,j,x COMBINATORIAL ANABELIAN TOPICS II 59 statement of assertion (v). Now the remainder of assertion (v) follows immediately from the various definitions involved [cf. also the situa- tion discussed in [CmbCsp], Definition 1.8, Proposition 1.9, Corollary 1.10, as well as the discussion, concerning specialization isomorphisms, preceding [CmbCsp], Definition 2.1; [CbTpI], Remark 5.6.1]. This com- pletes the proof of Lemma 3.6.  Definition 3.7. Let E ⊆ {1, · · · , n}. (i) Let v be an E-tripod of X n log [cf. Definition 3.1, (v)]; thus, v belongs to Vert(G i∈E,x ) for some choice of i ∈ E and x ∈ X n (k). Let j ∈ E \ {i} and e ∈ Edge(G j∈E\{i},x ). Then we shall say that v, or equivalently, an E-tripod of Π n associated to v [cf. Definition 3.3, (i)], arises from e if e = z i,j,x [cf. the statement new [cf. Lemma 3.6, (iv)]. of Lemma 3.6], and v = v i,j,x (ii) Let i ∈ E. Then we shall say that an E-tripod of Π n is i-central if #E = 3, and, moreover, the tripod is a verticial subgroup of the sort discussed in Lemma 3.6, (v), i.e., the unique, up to Π E -conjugacy, trigonal E-tripod of Π n contained in Π E/(E\{i}) [cf. the final portion of Lemma 3.6, (iv)]. We shall say that an E-tripod of Π n is central if it is j-central for some j ∈ E. Remark 3.7.1. Let E ⊆ {1, · · · , n}; T ⊆ Π E an E-tripod of Π n [cf. Definition 3.3, (i)]; σ ∈ S n ⊆ Out(Π n ) [cf. the discussion at the beginning of the present §3]; σ  ∈ Aut(Π n ) a lifting of σ ∈ S n ⊆ Out(Π n ). Write T σ  ⊆ Π σ(E) ∼ for the image of T ⊆ Π E by the isomorphism Π E → Π σ(E) determined by σ  ∈ Aut(Π n ). (i) One verifies easily that T σ  ⊆ Π σ(E) is a σ(E)-tripod of Π n . (ii) If, moreover, the equality #E = 3 holds, and T is i-central [cf. Definition 3.7, (ii)] for some i ∈ E, then one verifies easily from Lemma 3.6, (v), that T σ  ⊆ Π σ(E) is σ(i)-central. (iii) In the situation of (ii), let T  ⊆ Π E be a central E-tripod of Π n . Then it follows from Lemma 3.6, (v), that there exist an element τ ∈ S n ⊆ Out(Π n ) and a lifting τ  ∈ Aut(Π n ) of τ such that τ preserves the subset E ⊆ {1, · · · , n}, and, moreover, the image of T ⊆ Π E by the automorphism of Π E determined by τ  ∈ Aut(Π n ) coincides with T  ⊆ Π E . 60 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Lemma 3.8 (Strict tripods). Let E ⊆ {1, · · · , n} and T ⊆ Π E an E- tripod of Π n [cf. Definition 3.3, (i)] that arises as a verticial subgroup associated to a vertex v ∈ Vert(G i∈E,x ) for some i ∈ {1, · · · , n} [which thus implies that T ⊆ Π E/(E\{i}) ⊆ Π E ]. Then the following hold: (i) There exists a [not necessarily unique!] subset E  ⊆ E such that T is E  -strict [cf. Definition 3.3, (iii)]. In this situation,  i ∈ E, and, moreover, p Π E/E  : Π E  Π E induces an isomor- ∼ phism T → T E  onto an E  -tripod T E  of Π n . (ii) T is E-strict if and only if one of the following conditions is satisfied: (1) #E = 1. (2 C ) #E = 2; T ⊆ Π E is a verticial subgroup of Π E associated new ∈ Vert(G i∈E,x ) of Lemma 3.6, (iv), for to the vertex v i,j,x some choice of (i, j, x) such that z i,j,x ∈ Cusp(G j∈E\{i},x ). [In particular, T arises from z i,j,x ∈ Cusp(G j∈E\{i},x ) — cf. Definition 3.7, (i).] (2 N ) #E = 2; T ⊆ Π E is a verticial subgroup of Π E associated new ∈ Vert(G i∈E,x ) of Lemma 3.6, (iv), for to the vertex v i,j,x some choice of (i, j, x) such that z i,j,x ∈ Node(G j∈E\{i},x ). [In particular, T arises from z i,j,x ∈ Node(G j∈E\{i},x ) — cf. Definition 3.7, (i).] (3) #E = 3, and T is central [cf. Definition 3.7, (ii)]. (iii) Suppose that T is trigonal [cf. Definition 3.3, (ii)]. Then there exists a [not necessarily unique!] subset E  ⊆ E such that #E  = 3, and, moreover, the image of T ⊆ Π E via  p Π E/E  : Π E  Π E is a central tripod. Proof. Assertion (i) follows immediately from the various definitions involved by induction on #E , together with the well-known elemen- tary fact that any surjective endomorphism of a topologically finitely generated profinite group is necessarily bijective. Next, we verify as- sertion (ii). First, let us observe that sufficiency is immediate. Thus, it remains to verify necessity. Suppose that T is E-strict. Now one verifies easily that if there exists an element j ∈ E \ {i} such that c diag i,j,x ∈ C(v) [cf. Lemma 3.6, (ii)], then it follows immediately that the image of T ⊆ Π E via p Π E/(E\{j}) : Π E  Π E\{j} is an (E \ {j})-tripod [cf. also Lemma 3.6, (iii), (iv)]. Thus, since T is E-strict, we conclude that every cusp of G i∈E,x that is ∈ C(v) is non-diagonal. In particular, since v is of type (0, 3), it follows immediately from Lemma 3.2, (ii), that 0 ≤ #E − 1 ≤ #C(v) ≤ 3. If #C(v) = 0, then it follows from the inequality #E −1 ≤ #C(v) that #E = 1, i.e., condition (1) is satisfied. If #C(v) = 3, then one verifies easily that #E = 1, i.e., condition (1) COMBINATORIAL ANABELIAN TOPICS II 61 is satisfied. Thus, it remains to verify assertion (ii) in the case where #C(v) ∈ {1, 2}. Suppose that #C(v) = 1 and #E  = 1. Then it follows immediately from the inequality #E−1 ≤ #C(v) that #E = 2. Now let us recall [cf. Lemma 3.2, (ii)] that the number of diagonal cusps of G i∈E,x is = #E − 1 = 1. Moreover, the unique cusp on v is the unique diagonal cusp of G i∈E,x [cf. the argument of the preceding paragraph]. Thus, one verifies easily that T satisfies condition (2 N ). Next, suppose that #C(v) = 2 and #E  = 1. Then it follows immediately from the inequality #E−1 ≤ #C(v) that #E ∈ {2, 3}. Now let us recall [cf. Lemma 3.2, (ii)] that if #E = 2 (respectively, #E = 3), then the number of diagonal cusps of G i∈E,x is = #E−1, i.e., 1 (respectively, 2). Moreover, the set of diagonal cusp(s) of G i∈E,x is contained in (respectively, is equal to) C(v) [cf. the argument of the preceding paragraph]. Thus, one verifies easily that T satisfies condition (2 C ) (respectively, (3)). This completes the proof of assertion (ii). Finally, we verify assertion (iii). It follows from assertion (i) that there exists a subset E  ⊆ E such that T is E  -strict. Moreover, it follows immediately from the definition of a trigonal tripod that the  E  -tripod given by the image p Π E/E  (T ) ⊆ Π E is trigonal. On the other hand, if the E  -tripod p Π E/E  (T ) satisfies any of conditions (1), (2 C ), (2 N ) of assertion (ii), then one verifies easily that p Π E/E  (T ) is not trigonal [cf. Π the final portion of Lemma 3.6, (iv)]. Thus, p E/E  (T ) satisfies condition (3) of assertion (ii); in particular, p Π E/E  (T ) is central. This completes the proof of assertion (iii).  Lemma 3.9 (Generalities on normalizers and commensura- tors). Let G be a profinite group, N ⊆ G a normal closed subgroup of G, and H ⊆ G a closed subgroup of G. Then the following hold: (i) It holds that C G (H) ⊆ C G (H ∩ N ). (ii) It holds that C G (H) ⊆ N G (Z G loc (H)) [cf. the discussion entitled “Topological groups” in “Notations and Conventions”]. (iii) Suppose that H ⊆ N . Then it holds that C G (H) ⊆ N G (C N (H)). In particular, if, moreover, H is commensurably terminal in N , then it holds that C G (H) = N G (H). def def (iv) Write H = H/(H ∩ N ) ⊆ G = G/N . If H ∩ N is commen- surably terminal in N , and the image of C G (H) ⊆ G in G is contained in N G (H), then C G (H) = N G (H). Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii). Let g ∈ C G (H) and a ∈ Z G loc (H). Since Z G loc (H) = Z G loc (H ∩ (g −1 · H · g)) = Z G loc (g −1 · H · g), 62 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI there exists an open subgroup U ⊆ H of H such that a ∈ Z G (g −1 ·U ·g). But this implies that gag −1 ∈ Z G (U ) ⊆ Z G loc (H). This completes the proof of assertion (ii). Next, we verify assertion (iii). Let g ∈ C G (H) and a ∈ C N (H). Since C N (H) ⊆ C G (H) = C G (H ∩ (g −1 · H · g)) = C G (g −1 · H · g), we conclude that ag −1 · H · ga −1 is commensurate with g −1 · H · g. In particular, gag −1 · H · ga −1 g −1 is commensurate with H, i.e., gag −1 ∈ C G (H) ∩ N = C N (H). This completes the proof of assertion (iii). Finally, we verify assertion (iv). First, we observe that since H ∩ N is commensurably terminal in N , one verifies easily that H = N H·N (H ∩ N ). Let g ∈ C G (H). Then since the image of C G (H) ⊆ G in G is contained in N G (H), it is immediate that g · H · g −1 ⊆ H · N . On the other hand, again by applying the fact that H ∩ N is commensurably terminal in N , we conclude immediately from assertions (i), (iii), that C G (H) ⊆ C G (H ∩ N ) = N G (H ∩ N ). Thus, we obtain that (g · H · g −1 ) ∩ N = H ∩ N ; in particular, g · H · g −1 ⊆ N H·N ((g · H · g −1 ) ∩ N ) = N H·N (H ∩ N ) = H, i.e., g ∈ N G (H). This completes the proof of assertion (iv).  Lemma 3.10 (Restrictions of outomorphisms). Let G be a profi- nite group and H ⊆ G a closed subgroup of G. Write Out H (G) ⊆ Out(G) for the group of outomorphisms of G that preserve the G- conjugacy class of H. Suppose that the homomorphism N G (H) → Aut(H) determined by conjugation factors through Inn(H) ⊆ Aut(H). Then the following hold: (i) For α ∈ Out H (G), let us write α| H for the outomorphism of H determined by the restriction to H ⊆ G of a lifting α  ∈ Aut(G) of α such that α  (H) = H. Then α| H does not depend on the choice of the lifting “ α ”, and the map Out H (G) −→ Out(H) given by assigning α → α| H is a group homomorphism. (ii) The homomorphism Out H (G) −→ Out(H) of (i) depends only on the G-conjugacy class of the closed def subgroup H ⊆ G, i.e., if we write H γ = γ · H · γ −1 for γ ∈ G, then the diagram Out H (G) −−−→ Out(H) ⏐ ⏐  γ Out H (G) −−−→ Out(H γ ) — where the upper (respectively, lower) horizontal arrow is the homomorphism given by mapping α → α| H (respectively, COMBINATORIAL ANABELIAN TOPICS II 63 α → α| H γ ), and the right-hand vertical arrow is the isomor- ∼ phism obtained by conjugation via the isomorphism H → H γ determined by conjugation by γ ∈ G — commutes. Proof. Assertion (i) follows immediately from our assumption that the homomorphism N G (H) → Aut(H) determined by conjugation factors through Inn(H) ⊆ Aut(H), together with the various definitions in- volved. Assertion (ii) follows immediately from the various definitions involved. This completes the proof of Lemma 3.10.  Lemma 3.11 (Commensurator of a tripod arising from an edge). In the notation of Lemma 3.6, suppose that (j, i) = (1, 2); E = {i, j}; z i,j,x ∈ Edge(G j∈E\{i},x ). [Thus, G j∈E\{i},x = G i∈E\{j},x = G; ∼ ∼ Π 2 = Π E ; Π 1 = Π {j} → Π G j∈E\{i},x = Π G ; Π 2/1 = Π E/(E\{i}) → Π G i∈E,x .] def def def def Π Write G 2/1 = G i∈E,x ; G 1\2 = G j∈E,x ; p Π 1\2 = p E/{2} : Π 2  Π {2} ; Π 1\2 = ∼ def def diag = c diag Ker(p Π i,j,x ∈ 1\2 ) = Π E/{2} → Π G 1\2 ; z x = z i,j,x ∈ Edge(G); c def def new new = v i,j,x ∈ Vert(G 2/1 ) Cusp(G 2/1 ) [cf. Lemma 3.6, (ii)]; v new = v 2/1 new [cf. Lemma 3.6, (iv)]; v 1\2 ∈ Vert(G 1\2 ) for the vertex that corresponds ∼ to v new ∈ Vert(G 2/1 ) via the natural bijection Vert(G 2/1 ) → Vert(G 1\2 ) induced by the automorphism of X E log determined by permuting the fac- tors labeled i, j; Y → X E for the base-change — by the morphism log X E → X {1} × k X {2} = X × k X determined by p log E/{1} and p E/{2} — of the geometric point of X {1} × k X {2} = X × k X determined by the geomet- ric points x {1} of X {1} = X and x {1} of X {2} = X of Definition 3.1, (i) [i.e., as opposed to the geometric point of X {1} × k X {2} = X × k X deter- mined by the geometric points x {1} of X {1} = X and x {2} of X {2} = X]; Y log for the log scheme obtained by equipping Y with the log structure induced by the log structure of X E log ; U ⊆ Y for the 2-interior of Y log [cf. [MzTa], Definition 5.1, (i)]; U log for the log scheme obtained by equipping U with the log structure induced by the log structure of X E log ; Π U for the maximal pro-Σ quotient of the kernel of the natural sur- jection π 1 (U log )  π 1 ((Spec k) log ). [Thus, one verifies easily that Y is isomorphic to P 1 k ; that the complement Y \U consists of three closed new new and v 1\2 correspond to the closed points of Y ; that the vertices v 2/1 irreducible subscheme Y ⊆ X E ; and that the point corresponding to the cusp c diag is contained in Y — cf. Lemma 3.6, (iv).] Let Π z x ⊆ Π 1 be an edge-like subgroup associated to z x ∈ Edge(G); Π c diag ⊆ Π 2/1 ∩ Π 1\2 a cuspidal subgroup associated to c diag ; Π v new ⊆ Π 2/1 a verticial sub- def new = Π v new ; group associated to v new that contains Π c diag ⊆ Π 2 ; Π v 2/1 new new ⊆ Π 1\2 a verticial subgroup associated to v Π v 1\2 1\2 that contains def def Π c diag ⊆ Π 2 . Write Π 2 | z x = Π 2 × Π 1 Π z x ⊆ Π 2 ; D c diag = N Π 2 (Π c diag ); 64 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI def def I v new | z x = Z Π 2 | zx (Π v new ) ⊆ D v new | z x = N Π 2 | zx (Π v new ). Then the follow- ing hold: (i) It holds that D c diag ∩ Π 2/1 = D c diag ∩ Π 1\2 = C Π 2 (Π c diag ) ∩ Π 2/1 = C Π 2 (Π c diag ) ∩ Π 1\2 = Π c diag . (ii) It holds that C Π 2 (Π c diag ) = D c diag . Π (iii) The surjections p Π 2/1 : Π 2  Π 1 , p 1\2 : Π 2  Π {2} determine ∼ ∼ isomorphisms D c diag /Π c diag → Π 1 , D c diag /Π c diag → Π {2} , re- spectively, such that the resulting composite outer isomorphism ∼ Π 1 → Π {2} is the identity outer isomorphism. (iv) The natural inclusions Π v new , I v new | z x → D v new | z x determine ∼ an isomorphism Π v new × I v new | z x → D v new | z x = C Π 2 | zx (Π v new ). Moreover, the composite I v new | z x → D v new | z x → Π z x is an iso- morphism. (v) It holds that C Π 2 (D v new | z x ) ⊆ C Π 2 (Π v new ). (vi) D v new | z x is commensurably terminal in Π 2 , i.e., it holds that D v new | z x = C Π 2 (D v new | z x ). (vii) It holds that Z Π 2 (Π v new ) = Z Π loc 2 (Π v new ) = I v new | z x . Moreover, these profinite groups are isomorphic to Z Σ [cf. the discussion entitled “Numbers” in [CbTpI], §0]. (viii) It holds that C Π 2 (Π v new ) = D v new | z x = Π v new × Z Π 2 (Π v new ). In particular, the equality C Π 2 (Π v new ) = N Π 2 (Π v new ) holds. (ix) It holds that Z(C Π 2 (Π v new )) = Z Π 2 (Π v new ). (x) It holds that new ) ∩ Π 2/1 = Π v new , C Π 2 (Π v 2/1 2/1 new ) ∩ Π 1\2 = Π v new , C Π 2 (Π v 1\2 1\2 new ) = C Π (Π v new ) . C Π 2 (Π v 2/1 2 1\2 Moreover, for suitable choices of basepoints of the log schemes U log and X E log , the natural morphism U log → X E log induces an ∼ new ) = C Π (Π v new ). isomorphism Π U → C Π 2 (Π v 2/1 2 1\2 Proof. First, we verify assertion (i). Now it is immediate that we have inclusions Π c diag ⊆ D c diag ⊆ C Π 2 (Π c diag ). In particular, since Π c diag is commensurably terminal in Π 2/1 and Π 1\2 [cf. [CmbGC], Proposition 1.2, (ii)], we obtain that Π c diag ⊆ D c diag ∩ Π 2/1 ⊆ C Π 2 (Π c diag ) ∩ Π 2/1 = C Π 2/1 (Π c diag ) = Π c diag ; Π c diag ⊆ D c diag ∩ Π 1\2 ⊆ C Π 2 (Π c diag ) ∩ Π 1\2 = C Π 1\2 (Π c diag ) = Π c diag . This completes the proof of assertion (i). As- sertions (ii), (iii) follow immediately from assertion (i), together with COMBINATORIAL ANABELIAN TOPICS II 65 p Π 2/1 the [easily verified] fact that the composites D c diag → Π 2  Π 1 and p Π 1\2 D c diag → Π 2  Π {2} are surjective. Next, we verify assertion (iv). It follows immediately from the vari- ous definitions involved — by considering a suitable stable log curve of type (g, r) over (Spec k) log and applying a suitable specialization iso- morphism [cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] — that, to verify assertion (iv), we may assume without loss of generality that Cusp(G) ∪ {z x } = Edge(G). Then, in light of the well-known local structure of X log in a neigh- borhood of the node or cusp corresponding to z x , one verifies easily ∼ that the outer action Π z x → Out(Π 2/1 ) → Out(Π G 2/1 ) arising from the natural exact sequence 1 −→ Π 2/1 −→ Π 2 | z x −→ Π z x −→ 1 is of SNN-type [cf. [NodNon], Definition 2.4, (iii)], hence, in partic- ular, that the composite I v new | z x → D v new | z x → Π z x is an isomor- phism. Thus, assertion (iv) follows immediately from [NodNon], Re- mark 2.7.1, together with the commensurable terminality of Π v new in Π 2/1 [cf. [CmbGC], Proposition 1.2, (ii)] and the fact that the compos- ite D v new | z x → Π 2 | z x  Π z x is surjective. This completes the proof of assertion (iv). Next, we verify assertion (v). It follows immediately from asser- tion (iv), together with the commensurable terminality of Π v new in Π 2/1 [cf. [CmbGC], Proposition 1.2, (ii)], that D v new | z x ∩ Π 2/1 = Π v new . Thus, since Π 2/1 is normal in Π 2 , assertion (v) follows immediately from Lemma 3.9, (i). This completes the proof of assertion (v). Next, we verify assertion (vi). Since the image of the composite p Π 2/1 D v new | z x → Π 2  Π 1 coincides with Π z x ⊆ Π 1 [cf. assertion (iv)], and Π z x ⊆ Π 1 is commensurably terminal in Π 1 [cf. [CmbGC], Proposition 1.2, (ii)], it follows immediately that C Π 2 (D v new | z x ) ⊆ Π 2 | z x . In partic- ular, it follows immediately from assertions (iv), (v) that D v new | z x ⊆ C Π 2 (D v new | z x ) ⊆ C Π 2 (Π v new ) ∩ Π 2 | z x = C Π 2 | zx (Π v new ) = D v new | z x . This completes the proof of assertion (vi). Next, we verify assertion (vii). It follows from assertion (iv) and [CmbGC], Remark 1.1.3, that I v new | z x is isomorphic to Z Σ . Moreover, it follows from the various definitions involved that we have inclusions I v new | z x ⊆ Z Π 2 (Π v new ) ⊆ Z Π loc 2 (Π v new ). Thus, to verify assertion (vii), it suffices to verify that Z Π loc 2 (Π v new ) ⊆ I v new | z x . To this end, let us ob- serve that it follows immediately from the final portion of Lemma 3.6, ∼ new ) ⊆ Π {2} → Π G is an edge-like subgroup (iv), that the image p Π 1\2 (Π v ∼ of Π {2} → Π G associated to z x ∈ Edge(G). Thus, since every edge- like subgroup is commensurably terminal [cf. [CmbGC], Proposition 66 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ loc new )) ⊆ Π {2} → Π G 1.2, (ii)], it follows that the image p Π 1\2 (Z Π 2 (Π v ∼ is contained in an edge-like subgroup of Π {2} → Π G associated to z x ∈ Edge(G). On the other hand, since Π c diag ⊆ Π v new , we have Z Π loc 2 (Π v new ) ⊆ Z Π loc 2 (Π c diag ) ⊆ C Π 2 (Π c diag ) = D c diag [cf. assertion (ii)]. In particular, it follows immediately from assertion (iii), together with the fact [cf. assertion (iv)] that I v new | z x ⊆ Z Π loc 2 (Π v new ) surjects onto loc new )) ⊆ Π 1 is Π z x [cf. also [NodNon], Lemma 1.5], that p Π 2/1 (Z Π 2 (Π v loc contained in Π z x ⊆ Π 1 , i.e., Z Π 2 (Π v new ) ⊆ Π 2 | z x . Thus, it follows im- mediately from assertion (iv), together with the slimness of Π v new [cf. [CmbGC], Remark 1.1.3], that Z Π loc 2 (Π v new ) ⊆ I v new | z x . This completes the proof of assertion (vii). Next, we verify assertion (viii). It follows from assertion (vii), to- gether with Lemma 3.9, (ii), that C Π 2 (Π v new ) ⊆ N Π 2 (I v new | z x ). In par- ticular, since D v new | z x is generated by Π v new , I v new | z x [cf. assertion (iv)], it follows immediately that (D v new | z x ⊆) C Π 2 (Π v new ) ⊆ C Π 2 (D v new | z x ). Thus, the first equality of assertion (viii) follows from assertion (vi); the second equality of assertion (viii) follows immediately from assertions (iv), (vii). This completes the proof of assertion (viii). Next, we verify assertion (ix). Let us recall from [CmbGC], Remark 1.1.3, that Π v new is slim. Thus, assertion (ix) follows from assertion (viii), together with the final portion of assertion (vii). This completes the proof of assertion (ix). Finally, we verify assertion (x). The first two equalities follow from [CmbGC], Proposition 1.2, (ii). Next, let us observe that since [it is im- mediate that] the automorphism of X E log determined by permuting the new new factors labeled i, j stabilizes U , but permutes v 2/1 and v 1\2 , one verifies immediately that, to verify assertion (x), it suffices to verify that, for suitable choices of basepoints of the log schemes U log and X E log , the nat- ∼ ural morphism U log → X E log induces an isomorphism Π U → C Π 2 (Π v new ) new new )). To this end, let us observe that since the vertex v (= C Π 2 (Π v 2/1 corresponds to the closed irreducible subscheme Y ⊆ X E [cf. the dis- cussion following the definition of Π U in the statement of Lemma 3.11], it follows immediately from the various definitions involved that, for suitable choices of basepoints of the log schemes U log and X E log , the natural morphism U log → X E log gives rise to a commutative diagram 1 −−−→ Π U/z x −−−→ ⏐ ⏐   Π U ⏐ ⏐  −−−→ Π z x −−−→ 1 1 −−−→ Π v new −−−→ D v new | z x −−−→ Π z x −−−→ 1 — where we write Π U/z x for the kernel of the natural surjection Π U  Π z x ; the horizontal sequences are exact; the exactness of the lower horizontal sequence follows from assertion (iv); the left-hand vertical arrow is an isomorphism. Thus, it follows from assertion (viii) that, COMBINATORIAL ANABELIAN TOPICS II 67 for suitable choices of basepoints of the log schemes U log and X E log , ∼ the natural morphism U log → X E log induces an isomorphism Π U → D v new | z x = C Π 2 (Π v new ), as desired. This completes the proof of assertion (x), hence also of Lemma 3.11.  The first item of the following result [i.e., Lemma 3.12, (i)] is, along with its proof, a routine generalization of [CmbCsp], Corollary 1.10, (ii). Lemma 3.12 (Commensurator of a tripod). Let E ⊆ {1, · · · , n} and T ⊆ Π E an E-tripod of Π n [cf. Definition 3.3, (i)]. Then the following hold: (i) It holds that C Π E (T ) = T ×Z Π E (T ). Thus, if an outomorphism α of Π E preserves the Π E -conjugacy class of T , then one may define α| T ∈ Out(T ) [cf. Lemma 3.10, (i)]. (ii) Suppose that n = #E = 3, and that T is central [cf. Defi- nition 3.7, (ii)]. Let T  ⊆ Π E = Π n be a central E-tripod of Π n . Then C Π n (T ) (respectively, N Π n (T ); Z Π n (T )) is a Π n - conjugate of C Π n (T  ) (respectively, N Π n (T  ); Z Π n (T  )). Proof. Let i ∈ E; x ∈ X n (k); v ∈ Vert(G i∈E,x ) be such that v is of type (0, 3), and, moreover, T is a verticial subgroup of Π E associated to v ∈ Vert(G i∈E,x ). [Thus, we have an inclusion T ⊆ Π E/(E\{i}) ⊆ Π E — cf. Definition 3.1, (iv).] First, we verify assertion (i). Since T ⊆ Π E/(E\{i}) ⊆ Π E , and T is commensurably terminal in Π E/(E\{i}) [cf. [CmbGC], Proposition 1.2, (ii)], it follows from Lemma 3.9, (iii), that C Π E (T ) = N Π E (T ). Thus, in light of the slimness of T [cf. [CmbGC], Remark 1.1.3], to ver- ify assertion (i), it suffices to verify that the natural outer action of N Π E (T ) on T is trivial. To this end, let E  ⊆ E be such that T is E  -strict [cf. Lemma 3.8, (i)]; write T E  ⊆ Π E  for the image of T via  p Π E/E  : Π E  Π E . Then it is immediate that the image of N Π E (T ) Π via p E/E  : Π E  Π E  is contained in N Π E  (T E  ), and that the natural surjection T  T E  is an isomorphism [cf. Lemma 3.8, (i)]. Thus, one verifies easily — by replacing E, T by E  , T E  , respectively — that, to verify that the natural outer action of N Π E (T ) on T is trivial, we may assume without loss of generality that T is E-strict. If T satisfies condition (1) of Lemma 3.8, (ii), then assertion (i) follows from the commensurable terminality of T in Π E [cf. [CmbGC], Proposition 1.2, (ii)]. If T satisfies either condition (2 C ) or condition (2 N ) of Lemma 3.8, (ii), then assertion (i) follows immediately from Lemma 3.11, (viii). If T satisfies condition (3) of Lemma 3.8, (ii), then one verifies easily from the various definitions involved — by considering a suitable stable log 68 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI curve of type (g, r) over (Spec k) log and applying a suitable specializa- tion isomorphism [cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] — that, to verify assertion (i), we may assume without loss of generality that Node(G) = ∅. Thus, assertion (i) follows immediately from [CmbCsp], Corollary 1.10, (ii). This completes the proof of assertion (i). Next, we verify assertion (ii). Let us recall from Remark 3.7.1, (iii), that there exist an element τ ∈ S 3 ⊆ Out(Π 3 ) [cf. the discussion at the beginning of the present §3] and a lifting τ  ∈ Aut(Π 3 ) of τ such that the image of T ⊆ Π 3 by the automorphism τ  ∈ Aut(Π 3 ) coincides with T  ⊆ Π 3 . Next, let us observe that one verifies easily that τ ∈ S 3 may be written as a product of transpositions in S 3 . Thus, in the remainder of the proof of assertion (ii), we may assume without loss of generality that τ is a transposition in S 3 . Moreover, in the remainder of the proof of assertion (ii), we may assume without loss of generality, by conjugating by a suitable element of S 3 , that τ is the transposition “(1, 2)” in S 3 . Thus, if, moreover, i = 3 [i.e., the E-tripod T is 3-central], then it follows from Lemma 3.6, (v), that T is a Π 3 - conjugate of T  , hence that C Π 3 (T ) (respectively, N Π 3 (T ); Z Π 3 (T )) is a Π 3 -conjugate of C Π 3 (T  ) (respectively, N Π 3 (T  ); Z Π 3 (T  )). In particular, in the remainder of the proof of assertion (ii), we may assume without loss of generality, by conjugating by τ ∈ S 3 if necessary, that i = 2, i.e., that the E-tripods T , T  are 2-central, 1-central, respectively. Next, let us observe that, in this situation, one verifies immedi- ately from the various definitions involved that there exists a natural identification between Π {1,2,3}/{3} and the “Π 2 ” that arises in the case log log where we take “X log ” to be the base-change of p log {3} : X {2,3} → X {3} log via a suitable morphism of log schemes (Spec k) log → X {3} . More- over, one also verifies immediately from the various definitions involved [cf. also Lemma 3.6, (v)] that this natural identification maps suitable Π 3 -conjugates of T , T  , respectively, bijectively onto the closed sub- new ”, “Π v new ” of the “Π 2 ” that appears in the statement of groups “Π v 2/1 1\2 Lemma 3.11. In particular, it follows from Lemma 3.11, (viii), (ix), (x), that the following assertions hold: (a) The following equalities hold: C Π {1,2,3}/{3} (T ) = T × Z Π {1,2,3}/{3} (T ), C Π {1,2,3}/{3} (T  ) = T  × Z Π {1,2,3}/{3} (T  ). (b) The following equalities hold: C Π {1,2,3}/{3} (T ) ∩ Π {1,2,3}/{1,3} = T, C Π {1,2,3}/{3} (T  ) ∩ Π {1,2,3}/{2,3} = T  . COMBINATORIAL ANABELIAN TOPICS II 69 (c) The subgroup C Π {1,2,3}/{3} (T ) (respectively, Z Π {1,2,3}/{3} (T )) is a Π {1,2,3}/{3} -conjugate of the subgroup C Π {1,2,3}/{3} (T  ) (respec- tively, Z Π {1,2,3}/{3} (T  )). In particular, it follows from (c) that, to verify assertion (ii), it suffices to verify the following assertion: Claim 3.12.A: The following equalities hold: C Π 3 (T ) = C Π 3 (C Π {1,2,3}/{3} (T )), C Π 3 (T  ) = C Π 3 (C Π {1,2,3}/{3} (T  )), N Π 3 (T ) = N Π 3 (C Π {1,2,3}/{3} (T )), N Π 3 (T  ) = N Π 3 (C Π {1,2,3}/{3} (T  )), Z Π 3 (T ) = Z Π 3 (C Π {1,2,3}/{3} (T )), Z Π 3 (T  ) = Z Π 3 (C Π {1,2,3}/{3} (T  )). First, we verify the first four equalities of Claim 3.12.A. Observe that since Π {1,2,3}/{3} is a normal closed subgroup of Π 3 and contains both T and T  , it follows from Lemma 3.9, (iii), that the inclusions N Π 3 (T ) ⊆ C Π 3 (T ) ⊆ N Π 3 (C Π {1,2,3}/{3} (T )) ⊆ C Π 3 (C Π {1,2,3}/{3} (T )), N Π 3 (T  ) ⊆ C Π 3 (T  ) ⊆ N Π 3 (C Π {1,2,3}/{3} (T  )) ⊆ C Π 3 (C Π {1,2,3}/{3} (T  )) hold. Moreover, by the normality of Π {1,2,3}/{1,3} and Π {1,2,3}/{2,3} in Π 3 , one verifies easily, by applying (b), that the inclusions N Π 3 (C Π {1,2,3}/{3} (T )) ⊆ N Π 3 (T ), C Π 3 (C Π {1,2,3}/{3} (T )) ⊆ C Π 3 (T ), N Π 3 (C Π {1,2,3}/{3} (T  )) ⊆ N Π 3 (T  ), C Π 3 (C Π {1,2,3}/{3} (T  )) ⊆ C Π 3 (T  ) hold. This completes the proof of the first four equalities of Claim 3.12.A. Finally, we verify the final two equalities of Claim 3.12.A. Let us first observe that the inclusions T ⊆ C Π {1,2,3}/{3} (T ), T  ⊆ C Π {1,2,3}/{3} (T  ) imply that Z Π 3 (C Π {1,2,3}/{3} (T )) ⊆ Z Π 3 (T ), Z Π 3 (C Π {1,2,3}/{3} (T  )) ⊆ Z Π 3 (T  ). Thus, it follows immediately from (a) that, to verify the final two equal- ities of Claim 3.12.A, it suffices to verify the following assertion: Claim 3.12.B: The following inclusions hold: Z Π 3 (T ) ⊆ Z Π 3 (Z Π {1,2,3}/{3} (T )), Z Π 3 (T  ) ⊆ Z Π 3 (Z Π {1,2,3}/{3} (T  )). First, let us observe that one verifies immediately from the various def- initions involved that the natural identification that appears in the dis- cussion preceding assertion (a) in the present proof of Lemma 3.12, (ii), determines a natural identification between Π {2,3}/{3} and the “Π 1 = Π {2} ” that arises in the case where we take “X log ” to be as in the discussion preceding assertion (a) in the present proof of Lemma 3.12, (ii). Thus, it follows immediately from the final portion of Lemma 3.6, (iv), that the image J T ⊆ Π {2,3}/{3} of T ⊆ Π {1,2,3}/{3} in Π {2,3}/{3} corresponds, via the natural identification just discussed, to an edge- like subgroup of “Π 1 = Π {2} ” associated to the edge z x ∈ Edge(G) that appears in the statement of Lemma 3.11. Moreover, it follows 70 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI immediately from (c) and Lemma 3.11, (iv), (vii), that the surjection Π {1,2,3}/{3}  Π {2,3}/{3} induces an isomorphism ∼ Π {1,2,3}/{3} ⊇ Z Π {1,2,3}/{3} (T ) → J Z ⊆ Π {2,3}/{3} — where the closed subgroup J Z ⊆ Π {2,3}/{3} corresponds, via the nat- ural identification just discussed, to an edge-like subgroup of “Π 1 = Π {2} ” associated to the edge z x ∈ Edge(G) that appears in the state- ment of Lemma 3.11. Thus, we conclude immediately from [CmbGC], Proposition 1.2, (ii), together with the various definitions involved, ∼ that J T = J Z ( ← Z Π {1,2,3}/{3} (T )). In particular, since Z Π 3 (T ) ⊆ N Π 3 (Z Π {1,2,3}/{3} (T )), and the surjection Π {1,2,3}/{3}  Π {2,3}/{3} in- duces a homomorphism Z Π 3 (T ) → Z Π {2,3}/{3} (J T ), one verifies easily that the first inclusion of Claim 3.12.B holds. The second inclusion of Claim 3.12.B follows from the first inclusion of Claim 3.12.B by applying τ  . This completes the proof of Claim 3.12.B, hence also of Lemma 3.12.  Lemma 3.13 (Preservation of verticial subgroups). In the nota- tion of Lemma 3.11, let α  be an F-admissible automorphism of Π E = Π 2 , v ∈ Vert(G). Write v ◦ ∈ Vert(G 2/1 ) for the vertex of G 2/1 that cor-  2/1 responds to v ∈ Vert(G) via the bijection of Lemma 3.6, (iv); α  1 , α for the automorphisms of Π 1 , Π 2/1 determined by α  ; α, α 1 , α 2/1 for the outomorphisms of Π 2 , Π 1 , Π 2/1 determined by α  , α  1 , α  2/1 , respectively. Then the following hold: ∼ (i) Recall the edge-like subgroup Π z x ⊆ Π 1 → Π G associated to the edge z x ∈ Edge(G). Suppose that α  1 (Π z x ) = Π z x . Suppose, moreover, either that ∼ (a) the outomorphism α 2/1 of Π G 2/1 ← Π 2/1 maps some cusp- ∼ idal inertia subgroup of Π G 2/1 ← Π 2/1 to a cuspidal inertia ∼ subgroup of Π G 2/1 ← Π 2/1 , or that (b) z x ∈ Cusp(G). [For example, condition (a) holds if the outomorphism α 2/1 of ∼ Π G 2/1 ← Π 2/1 is group-theoretically cuspidal — cf. [CmbGC], Definition 1.4, (iv).] Then α 2/1 preserves the Π 2/1 -conjugacy ∼ class of the verticial subgroup Π v new ⊆ Π 2/1 → Π G 2/1 associated to the vertex v new ∈ Vert(G 2/1 ). If, moreover, α 2/1 is group- theoretically cuspidal, then the induced outomorphism of Π v new [cf. Lemma 3.12, (i)] is itself group-theoretically cus- pidal. COMBINATORIAL ANABELIAN TOPICS II 71 (ii) In the situation of (i), suppose, moreover, that there exists a ∼ ∼ verticial subgroup Π v ⊆ Π G ← Π 1 of Π G ← Π 1 associated to v ∈ Vert(G) such that α  1 preserves the Π 1 -conjugacy class of Π v . Then α 2/1 preserves the Π 2/1 -conjugacy class of a ∼ verticial subgroup of Π G 2/1 ← Π 2/1 associated to the vertex v ◦ ∈ Vert(G 2/1 ). (iii) In the situation of (i), suppose, moreover, that X log is of type ∼ def (0, 3) [which implies that Π v = Π G ← Π 1 is the unique verti- cial subgroup of Π G associated to v], and that α 1 ∈ Out C (Π v ) cusp [cf. Definition 3.4, (i)]. Then there exists a geometric [cf. ∼ ∼ Definition 3.4, (ii)] outer isomorphism Π v new → Π v (= Π G ← Π 1 ) which satisfies the following condition: If either α 1 ∈ Out(Π 1 ) = Out(Π v ) is contained in Out(Π v ) Δ [cf. Definition 3.4, (i)] or α| Π v new ∈ Out(Π v new ) [cf. (i); Lemma 3.12, (i)] is contained in Out(Π v new ) Δ , then the outomorphisms α| Π v new , α 1 of Π v new , Π v are compatible relative to the outer ∼ isomorphism in question Π v new → Π v . def Proof. First, we verify assertions (i), (ii). Write S = Node(G 2/1 ) \ N (v new ). Then it follows immediately from the well-known local struc- ture of X log in a neighborhood of the edge corresponding to z x that if z x ∈ Node(G) (respectively, z x ∈ Cusp(G)), then the outer action of Π z x on Π (G 2/1 ) S [cf. [CbTpI], Definition 2.8] obtained by conjugating ∼ the natural outer action Π z x → Π 1 → Out(Π 2/1 ) → Out(Π G 2/1 ) — where the second arrow is the outer action determined by the exact sequence of profinite groups p Π 2/1 1 −→ Π 2/1 −→ Π 2 −→ Π 1 −→ 1 ∼ — by the natural outer isomorphism Φ (G 2/1 ) S : Π (G 2/1 ) S → Π G 2/1 [cf. [CbTpI], Definition 2.10] is of SNN-type [cf. [NodNon], Definition 2.4, (iii)] (respectively, IPSC-type [cf. [NodNon], Definition 2.4, (i)]). Thus, it follows immediately [in light of the various assumptions made in the statement of assertion (i)!] in the case of condition (a) (respectively, condition (b)) from Theorem 1.9, (i) (respectively, Theorem 1.9, (ii)), that the outomorphism α (G 2/1 ) S of Π (G 2/1 ) S obtained by conjugat- Φ (G ∼ 2/1 ) S ∼ ← Π (G 2/1 ) S is group- ing α 2/1 by the composite Π 2/1 → Π G 2/1 theoretically verticial [cf. [CmbGC], Definition 1.4, (iv)] and group- theoretically nodal [cf. [NodNon], Definition 1.12]. On the other hand, it follows immediately from condition (3) of [CbTpI], Proposition 2.9, ∼ (i), that the image via Φ (G 2/1 ) S : Π (G 2/1 ) S → Π G 2/1 of any verticial 72 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI subgroup of Π (G 2/1 ) S associated to the vertex of (G 2/1 ) S correspond- ing to v new is a verticial subgroup of Π G 2/1 associated to v new . Thus, since α (G 2/1 ) S is group-theoretically verticial, it follows immediately that α 2/1 preserves the Π 2/1 -conjugacy class of the verticial subgroup ∼ Π v new ⊆ Π 2/1 → Π G 2/1 associated to v new . [Here, we observe in passing the following easily verified fact: a vertex of (G 2/1 ) S corresponds to v new if and only if the verticial subgroup of Π (G 2/1 ) S associated to this p Π 1\2 ∼ vertex maps, via the composite Π (G 2/1 ) S → Π 2/1  Π {2} , to an abelian subgroup of Π {2} .] If, moreover, α 2/1 is group-theoretically cuspidal, then the group-theoretic cuspidality of the resulting outomorphism of Π v new follows immediately from the group-theoretic cuspidality of α 2/1 and the group-theoretic nodality of α (G 2/1 ) S . This completes the proof of assertion (i). To verify assertion (ii), let us first observe that it follows immedi- ately from [CbTpI], Theorem A, (i), that — after possibly replacing α  by the composite of α  with an inner automorphism of Π 2 determined by conjugation by an element of Π 2/1 — we may assume without loss of generality that if we write α  {2} for the automorphism of Π {2} deter- mined by α  , then α  {2} (Π v ) = Π v — where, by abuse of notation, we write Π v for some fixed subgroup of ∼ Π {2} whose image in Π G ← Π {2} is a verticial subgroup associated to v. ∼ Next, let us fix a verticial subgroup Π v ◦ ⊆ Π 2/1 → Π G 2/1 of Π G 2/1 as- sociated to the vertex v ◦ ∈ Vert(G 2/1 ) such that the composite Π v ◦ → p Π 1\2 ∼ Π 2/1  Π {2} determines an isomorphism Π v ◦ → Π v . Then let us ob- serve that one verifies easily from condition (3) of [CbTpI], Proposition 2.9, (i), together with [NodNon], Lemma 1.9, (ii), that there exists a unique vertex w ◦ ∈ Vert((G 2/1 ) S ) such that the image Π w ◦ ⊆ Π 2/1 Φ (G 2/1 ) S ∼ ∼ via the composite Π (G 2/1 ) S → Π G 2/1 ← Π 2/1 of some verticial ◦ subgroup of Π (G 2/1 ) S associated to w contains the verticial subgroup ∼ Π v ◦ ⊆ Π 2/1 → Π G 2/1 . Thus, it follows immediately from the vari- p Π 1\2 ous definitions involved that the composite Π w ◦ → Π 2/1  Π {2} is an injective homomorphism whose image Π w ⊆ Π {2} maps via the ∼ Φ G S ∼ def composite Π {2} → Π G ← Π G S — where we write S = Node(G) \ (Node(G) ∩ {z x }) — to a verticial subgroup of Π G S associated to a vertex w ∈ Vert(G S ). Here, we note that the vertex w may also be characterized as the unique vertex of G S such that the image via ∼ the natural outer isomorphism Φ G S : Π G S → Π G of some verticial subgroup associated to w contains a verticial subgroup associated to COMBINATORIAL ANABELIAN TOPICS II 73 ∼ v ∈ Vert(G). Thus, we obtain an isomorphism Π w ◦ → Π w , hence also ∼ an isomorphism α  2/1 (Π w ◦ ) → α  {2} (Π w ). Next, let us observe that since α (G 2/1 ) S is group-theoretically ver- ticial [cf. the argument given in the proof of assertion (i)], it fol- ∼ lows immediately that α  2/1 (Π w ◦ ) ⊆ Π 2/1 → Π (G 2/1 ) S is a verticial subgroup of Π (G 2/1 ) S that maps isomorphically to a verticial sub- ∼ group α  {2} (Π w ) ⊆ Π {2} → Π G S of Π G S that contains α  {2} (Π v ) = Π v . On the other hand, in light of the unique characterization of w ∼ given above, this implies that α  {2} (Π w ) ⊆ Π {2} → Π G S is a verti- cial subgroup associated to w, and hence [as is easily verified] that ∼ α  2/1 (Π w ◦ ) ⊆ Π 2/1 → Π (G 2/1 ) S is a verticial subgroup associated to w ◦ . In particular, one may apply the natural outer isomorphisms ∼ ∼  2/1 (Π w ◦ ); Π (G| H w ) Tw → α  {2} (Π w ) [cf. [CbTpI], Π ((G 2/1 )| H w ◦ ) Tw ◦ → α Definitions 2.2, (ii); 2.5, (ii)] arising from condition (3) of [CbTpI], Proposition 2.9, (i); moreover, one verifies easily that the resulting ∼ outer isomorphism Π ((G 2/1 )| H w ◦ ) Tw ◦ → Π (G| H w ) Tw [induced by the above ∼  {2} (Π w )] arises from scheme theory, hence isomorphism α  2/1 (Π w ◦ ) → α is graphic [cf. [CmbGC], Definition 1.4, (i)]. Therefore, we conclude ∼ that the closed subgroup α  2/1 (Π v ◦ ) ⊆ ( α 2/1 (Π w ◦ ) ⊆) Π 2/1 → Π G 2/1 is a verticial subgroup of Π G 2/1 associated to v ◦ . This completes the proof of assertion (ii). Finally, we verify assertion (iii). First, we recall from [CmbCsp], Corollary 1.14, (ii), that there exists an outer modular symmetry σ ∈ σ ∼ Π p 2/1 (S 5 ⊆) Out(Π 2 ) such that the composite Π v new → Π 2 → Π 2  Π 1 = Π v determines a(n) [necessarily geometric] outer isomorphism ∼ Π v new → Π v . The remainder of the proof of assertion (iii) is devoted to ∼ verifying that this outer isomorphism Π v new → Π v satisfies the condi- tion of assertion (iii). First, suppose that α 1 ∈ Out(Π 1 ) Δ . Then since Out F (Π 2 ) = Out FC (Π 2 ) = Out FCP (Π 2 ) [cf. [CmbCsp], Definition 1.1, (iv); Theorem 2.3, (ii), (iv), of the present monograph; our assumption that X log is of type (0, 3)], it follows from [CmbCsp], Corollary 1.14, (i), together with the injectivity portion of [CmbCsp], Theorem A, (i), that α commutes with every modular outer symmetry on Π 2 ; in particular, α commutes with σ. Thus, it follows immediately from [CmbCsp], Corol- ∼ lary 1.14, (iii), that the above outer isomorphism Π v new → Π v satisfies the condition of assertion (iii). def Next, suppose that α| Π v new ∈ Out(Π v new ) Δ . If we write α σ = σ ◦ α ◦ σ −1 (∈ Out FC (Π 2 ) cusp — cf. [CmbCsp], Corollary 1.14, (i); Theo- rem 2.3, (ii), and Lemma 3.5 of the present monograph) and (α σ ) 1 ∈ Out(Π v ) for the outomorphism of Π v determined by α σ , then it fol- lows immediately from [CmbCsp], Corollary 1.14, (iii), that the out- omorphisms α| Π v new , (α σ ) 1 of Π v new , Π v are compatible relative to the 74 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ outer isomorphism Π v new → Π v discussed above. Thus, since α| Π v new ∈ Out(Π v new ) Δ , we conclude that (α σ ) 1 ∈ Out(Π v ) Δ . In particular, [since Out F (Π 2 ) = Out FC (Π 2 ) = Out FCP (Π 2 ) — cf. [CmbCsp], Definition 1.1, (iv); Theorem 2.3, (ii), (iv), of the present monograph; our assumption that X log is of type (0, 3)] it follows from [CmbCsp], Corollary 1.14, (i), together with the injectivity portion of [CmbCsp], Theorem A, (i), that α σ commutes with every modular outer symmetry on Π 2 . Thus, we conclude that α σ commutes with σ −1 , which implies that α = α σ . This completes the proof of assertion (iii).  Lemma 3.14 (Commensurator of the closed subgroup arising from a certain second log configuration space). Let i ∈ E, j ∈ E, x, and z i,j,x be as in Lemma 3.6; let v ∈ Vert(G j∈E\{i},x ). Then, by ap- plying a similar argument to the argument used in [CmbCsp], Definition 2.1, (iii), (vi), or [NodNon], Definition 5.1, (ix), (x) [i.e., by consider- ing the portion of the underlying scheme X E of X E log corresponding to the underlying scheme (X v ) 2 of the 2-nd log configuration space (X v ) log 2 of the stable log curve X v log determined by G j∈E\{i},x | v — cf. [CbTpI], Definition 2.1, (iii)], one obtains a closed subgroup (Π v ) 2 ⊆ Π E/(E\{i,j}) [which is well-defined up to Π E -conjugation]. Write def (Π v ) 2/1 = (Π v ) 2 ∩ Π E/(E\{i}) ⊆ (Π v ) 2 . [Thus, one verifies easily that there exists a natural commutative dia- gram 1 −−−→ (Π v ) 2/1 ⏐ ⏐  −−−→ (Π v ) 2 ⏐ ⏐  −−−→ Π v ⏐ ⏐  −−−→ 1 p Π E/(E\{i}) 1 −−−→ Π E/(E\{i}) −−−→ Π E/(E\{i,j}) −−−−−−→ Π (E\{i})/(E\{i,j}) −−−→ 1 — where we use the notation Π v to denote a verticial subgroup of ∼ Π G j∈E\{i},x ← Π (E\{i})/(E\{i,j}) associated to v ∈ Vert(G j∈E\{i},x ), the horizontal sequences are exact, and the vertical arrows are injective.] Then the following hold: (i) Suppose that z i,j,x ∈ VCN(G j∈E\{i},x ) is contained in E(v). Write v ◦ ∈ Vert(G i∈E,x ) for the vertex of G i∈E,x that corre- sponds to v ∈ Vert(G j∈E\{i},x ) via the bijections of Lemma 3.6, ∼ new (i), (iv). Let Π v ◦ , Π v i,j,x ⊆ Π G i∈E,x ← Π E/(E\{i}) be verti- ∼ cial subgroups of Π G i∈E,x ← Π E/(E\{i}) associated to the ver- new new ⊆ tices v ◦ , v i,j,x ∈ Vert(G i∈E,x ), respectively, such that Π v i,j,x new (Π v ) 2/1 , and, moreover, Π v ◦ ∩ Π v i,j,x  = {1}. Let us say that two COMBINATORIAL ANABELIAN TOPICS II 75 Π E/(E\{i}) -conjugates Π γv ◦ , Π δv i,j,x new [i.e., where γ, δ ∈ Π E/(E\{i}) ] γ δ new are conjugate-adjacent if Π ◦ ∩ Π new  = {1}. of Π v ◦ , Π v i,j,x v v i,j,x Let us say that a finite sequence of Π E/(E\{i}) -conjugates of Π v ◦ , new is a conjugate-chain if any two adjacent members of Π v i,j,x the finite sequence are conjugate-adjacent. Let us say that a subgroup of Π E/(E\{i}) is conjugate-tempered if it appears as the first member of a conjugate-chain whose final mem- new . Then (Π v ) 2/1 is equal to the subgroup ber is equal to Π v i,j,x of Π E/(E\{i}) topologically generated by the conjugate-tempered subgroups and the elements δ ∈ Π E/(E\{i}) such that Π δv i,j,x new is conjugate-tempered. (ii) If N Π E\{i} (Π v ) = C Π E\{i} (Π v ), then N Π E ((Π v ) 2 ) = C Π E ((Π v ) 2 ). (iii) If C Π E\{i} (Π v ) = Π v × Z Π E\{i} (Π v ), then C Π E ((Π v ) 2 ) = (Π v ) 2 × Z Π E ((Π v ) 2 ). (iv) Suppose that v is of type (0, 3), i.e., that Π v is an (E \ {i})-tripod of Π n [cf. Definition 3.3, (i)]. Then it holds that C Π E ((Π v ) 2 ) = (Π v ) 2 × Z Π E ((Π v ) 2 ). Thus, if an outomorphism α of Π E preserves the Π E -conjugacy class of (Π v ) 2 , then one may define α| (Π v ) 2 ∈ Out((Π v ) 2 ) [cf. Lemma 3.10, (i)]. Proof. First, we verify assertion (i). We begin by observing that it follows immediately from [NodNon], Lemma 1.9, (ii), together with new ⊆ Π E/(E\{i}) [cf. the commensurable terminality of Π v i,j,x [CmbGC], Proposition 1.2, (ii)], that the subgroup described in the final portion of the statement of assertion (i) is contained in (Π v ) 2/1 . If #(N (v ◦ ) ∩ new N (v i,j,x )) = 1, then assertion (i) follows immediately from a similar argument to the argument applied in the proof of [CmbCsp], Propo- sition 1.5, (iii), together with the various definitions involved [cf. also [NodNon], Lemma 1.9, (ii)]. Thus, we may assume without loss of new )) = 2. generality that #(N (v ◦ ) ∩ N (v i,j,x Write new • e 1 ∈ N (v ◦ ) ∩ N (v i,j,x ) for the [uniquely determined — cf. new ( = {1}) [NodNon], Lemma 1.5] node such that Π v ◦ ∩ Π v i,j,x is a nodal subgroup associated to e 1 [cf. [NodNon], Lemma 1.9, (i)]; new • e 2 for the unique element of N (v ◦ ) ∩ N (v i,j,x ) such that e 2  = e 1 ◦ new [so N (v ) ∩ N (v i,j,x ) = {e 1 , e 2 }]; • H for the sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of the underlying semi-graph of G i∈E,x whose set of new vertices = {v ◦ , v i,j,x }; def • S = Node(G i∈E,x | H ) \ {e 1 , e 2 } [cf. [CbTpI], Definition 2.2, (ii)]; 76 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI def • H = (G i∈E,x | H ) S [which is well-defined since, as is easily ver- ified, S is not of separating type as a subset of Node(G i∈E,x | H ) — cf. [CbTpI], Definition 2.5, (i), (ii)]. Then it follows immediately from the construction of H that H {e 1 } [cf. [CbTpI], Definition 2.8], where we observe that one verifies eas- ily that the node e 1 of G i∈E,x may be regarded as a node of H, is cyclically primitive [cf. [CbTpI], Definition 4.1]. Moreover, it follows immediately from [NodNon], Lemma 1.9, (ii), together with the vari- ∼ ous definitions involved, that (Π v ) 2/1 ⊆ Π E/(E\{i}) → Π G i∈E,x may be characterized uniquely as the closed subgroup of Π G i∈E,x that contains new ⊆ Π G Π v i,j,x and, moreover, belongs to the Π G i∈E,x -conjugacy class i∈E,x of closed subgroups of Π G i∈E,x obtained by forming the image of the composite of outer homomorphisms Π H {e 1 } Φ H {e } 1 ∼ → Π H → Π G i∈E,x [cf. [CbTpI], Definition 2.10] — where the second arrow is the outer in- jection discussed in [CbTpI], Proposition 2.11. In particular, it follows from the commensurable terminality of (Π v ) 2/1 in Π G i∈E,x [cf. [CmbGC], Proposition 1.2, (ii)] that this characterization of (Π v ) 2/1 determines an ∼ outer isomorphism Π H {e 1 } → (Π v ) 2/1 . On the other hand, it follows immediately from a similar argument to the argument applied in the proof of [CmbCsp], Proposition 1.5, (iii), together with the various definitions involved [cf. also [NodNon], Lemma 1.9, (ii)], that the image of the closed subgroup of (Π v ) 2/1 topo- ∼ new via the inverse (Π v ) 2/1 → Π H logically generated by Π v ◦ and Π v i,j,x {e 1 } of this outer isomorphism is a verticial subgroup of Π H {e 1 } associated to the unique vertex of H {e 1 } . Thus, since H {e 1 } is cyclically primi- tive, assertion (i) follows immediately from [CmbGC], Proposition 1.2, (ii); [NodNon], Lemma 1.9, (ii), together with the description of the structure of a certain tempered covering of H {e 1 } given in [CbTpI], Lemma 4.3. This completes the proof of assertion (i). Next, we verify assertion (ii). Since (Π v ) 2/1 = (Π v ) 2 ∩ Π E/(E\{i}) is commensurably terminal in Π E/(E\{i}) [cf. [CmbGC], Proposition 1.2, (ii)], assertion (ii) follows immediately from Lemma 3.9, (iv). This completes the proof of assertion (ii). Next, we verify assertion (iii). First, let us observe that if E(v) = ∅, then one verifies immediately that the vertical arrows of the commutative diagram in the statement of Lemma 3.14 are isomorphisms, and hence that assertion (iii) holds. Thus, we may assume that E(v)  = ∅. Next, let us observe that it follows from assertion (ii) that N Π E ((Π v ) 2 ) = C Π E ((Π v ) 2 ). Thus, in light of the slimness of (Π v ) 2 [cf. [MzTa], Proposition 2.2, (ii)], to verify assertion (iii), it suffices to verify that the natural outer action of N Π E ((Π v ) 2 ) on (Π v ) 2 is trivial. On the other hand, since [one verifies easily that] COMBINATORIAL ANABELIAN TOPICS II 77 the natural outer action N Π E ((Π v ) 2 ) → Out((Π v ) 2 ) factors through Out F ((Π v ) 2 ) ⊆ Out((Π v ) 2 ), it follows from the injectivity portion of Theorem 2.3, (i) [cf. our assumption that E(v)  = ∅], that to verify the triviality in question, it suffices to verify that the natural outer action of N Π E ((Π v ) 2 ) on Π v is trivial. But this follows from the equality C Π E\{i} (Π v ) = Π v × Z Π E\{i} (Π v ). This completes the proof of assertion (iii). Assertion (iv) follows immediately from assertion (iii), together with Lemma 3.12, (i). This completes the proof of Lemma 3.14.  Lemma 3.15 (Preservation of various subgroups of geomet- ric origin). In the notation of Lemma 3.14, let α  be an F-admissible automorphism of Π E . Write α  E\{i} , α  E/(E\{i}) for the automorphisms  ; α, α E\{i} , α E/(E\{i}) for the outo- of Π E\{i} , Π E/(E\{i}) determined by α  , α  E\{i} , α  E/(E\{i}) , morphisms of Π E , Π E\{i} , Π E/(E\{i}) determined by α respectively. Suppose that there exist an edge e ∈ Edge(G j∈E\{i},x ) of G j∈E\{i},x that belongs to E(v) ⊆ Edge(G j∈E\{i},x ) and a pair Π e ⊆ ∼ Π v ⊆ Π G j∈E\{i},x ← Π (E\{i})/(E\{i,j}) of VCN-subgroups associated to e ∈ Edge(G j∈E\{i},x ), v ∈ Vert(G j∈E\{i},x ), respectively, such that α  E\{i} (Π e ) = Π e ⊆ α  E\{i} (Π v ) = Π v . Suppose, moreover, either that ∼ (a) the outomorphism α E/(E\{i}) of Π G i∈E,x ← Π E/(E\{i}) maps some ∼ cuspidal inertia subgroup of Π G i∈E,x ← Π E/(E\{i}) to a cuspidal ∼ inertia subgroup of Π G i∈E,x ← Π E/(E\{i}) , or that (b) e ∈ Cusp(G j∈E\{i},x ). [For example, condition (a) holds if the outomorphism α E/(E\{i}) of ∼ Π G i∈E,x ← Π E/(E\{i}) is group-theoretically cuspidal — cf. [CmbGC], Definition 1.4, (iv).] Write T ⊆ Π E for the E-tripod of Π n [cf. Def- inition 3.3, (i)] arising from e ∈ Edge(G j∈E\{i},x ) [cf. Definition 3.7, (i)]. Then the following hold: (i) The outomorphism α preserves the Π E -conjugacy classes of T , (Π v ) 2 ⊆ Π E . If, moreover, the outomorphism α E/(E\{i}) ∼ of Π G i∈E,x ← Π E/(E\{i}) is group-theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)], then the outomorphism α| T [cf. Lemma 3.12, (i)] of T is contained in Out C (T ) cusp [cf. Definition 3.4, (i)]. (ii) Suppose, moreover, that v is of type (0, 3) — i.e., that Π v is an (E\{i})-tripod of Π n — and that α E\{i} | Π v ∈ Out C (Π v ) cusp [cf. Lemma 3.12, (i)]. Then there exists a geometric [cf. Def- ∼ inition 3.4, (ii)] outer isomorphism T → Π v which satisfies the following condition: 78 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI If either α| T ∈ Out(T ) Δ [cf. (i)] or α E\{i} | Π v ∈ Out(Π v ) Δ , then the outomorphisms α| T , α E\{i} | Π v of T , Π v are compatible relative to the outer isomor- ∼ phism in question T → Π v . If, moreover, Π v is (E \ {i})-strict [cf. Definition 3.3, (iii)], then the following hold: (1) If #(E\{i}) = 1 [i.e., Π v satisfies condition (1) of Lemma 3.8, (ii)], then T is E-strict [i.e., T satisfies one of the two conditions (2 C ), (2 N ) of Lemma 3.8, (ii)]. (2) If #(E \ {i}) = 2 [i.e., Π v satisfies one of the two con- ditions (2 C ), (2 N ) of Lemma 3.8, (ii)], and the edge e ∈ Edge(G j∈E\{i},x ) is the unique diagonal cusp of G j∈E\{i},x [cf. Lemma 3.2, (ii)], then T is E-strict [i.e., T satisfies condition (3) of Lemma 3.8, (ii)], hence also central [cf. Definition 3.7, (ii)]. Proof. First, let us observe that one verifies easily — by replacing x by a suitable k-valued geometric point of X n (k) that lifts x E\{i,j} ∈ X E\{i,j} (k) [note that this does not affect “G j∈E\{i},x ”!] — that, to verify Lemma 3.15, we may assume without loss of generality that z i,j,x = e ∈ Edge(G j∈E\{i},x ). Now we verify assertion (i). First, let us observe that one verifies eas- log log ily — by replacing X E log by the base-change of p log E\{i,j} : X E → X E\{i,j} log by a suitable morphism of log schemes (Spec k) log → X E\{i,j} that lies over x E\{i,j} ∈ X E\{i,j} (k) [cf. Definition 3.1, (i)] — that, to verify asser- tion (i), we may assume without loss of generality that #E = 2. Then it follows immediately from Lemma 3.13, (i), that α E/(E\{i}) preserves the new ) ⊆ Π E/(E\{i}) . Moreover, it fol- Π E/(E\{i}) -conjugacy class of T (= Π v i,j,x lows immediately from Lemma 3.13, (i), (ii), together with Lemma 3.14, (i), that α E/(E\{i}) preserves the Π E/(E\{i}) -conjugacy classes of the normally terminal closed subgroups Π v ◦ ⊆ (Π v ) 2/1 ⊆ Π E/(E\{i}) [cf. [CmbGC], Proposition 1.2, (ii)]. In particular, since α  E\{i} (Π v ) = Π v , ∼ out by considering the natural isomorphism (Π v ) 2 → (Π v ) 2/1  Π v [cf. the upper exact sequence of the commutative diagram in the state- ment of Lemma 3.14; the discussion entitled “Topological groups” in [CbTpI], §0], we conclude that α E preserves the Π E -conjugacy class of (Π v ) 2 ⊆ Π E . ∼ Next, suppose that the outomorphism α E/(E\{i}) of Π G i∈E,x ← Π E/(E\{i}) is group-theoretically cuspidal. Then it follows from Lemma 3.13, (i), that α| T ∈ Out C (T ). Moreover, since α E/(E\{i}) is group-theoretically cuspidal, it follows immediately from Lemma 3.2, (iv), that α E/(E\{i}) fixes the Π E/(E\{i}) -conjugacy class of cuspidal inertia subgroups asso- new ) ( c diag ciated to each element ∈ C(v i,j,x i,j,x ). Thus, to verify that α| T ∈ COMBINATORIAL ANABELIAN TOPICS II 79 Out C (T ) cusp , it suffices to verify that α E/(E\{i}) fixes the Π E/(E\{i}) - ∼ conjugacy class of nodal subgroups of Π G i∈E,x ← Π E/(E\{i}) associated to new new each element of N (v i,j,x )∩N (v ◦ ). To this end, let e ◦ ∈ N (v i,j,x )∩N (v ◦ ) ∼ and Π e ◦ ⊆ Π G i∈E,x ← Π E/(E\{i}) a nodal subgroup associated to the node e ◦ such that Π e ◦ ⊆ Π v ◦ . Now let us observe that one verifies ∼ easily that the closed subgroups Π e ◦ ⊆ Π v ◦ ⊆ Π G i∈E,x ← Π E/(E\{i}) ∼ map bijectively onto VCN-subgroups of Π G i∈E\{j} ,x ← Π (E\{j})/(E\{i,j}) associated, respectively, to the edge and vertex of G i∈E\{j},x that corre- spond, via the bijections of Lemma 3.6, (i), to e, v ∈ VCN(G j∈E\{i},x ). In particular, if β  is the composite of α  with some Π E/(E\{i}) -inner  v ◦ ) = Π v ◦ [cf. the preceding paragraph], automorphism such that β(Π then it follows immediately from our assumption that α  E\{i} (Π e ) = Π e ⊆ α  E\{i} (Π v ) = Π v , together with [CbTpI], Theorem A, (i), and [CmbGC], Proposition 1.2, (ii), that the automorphism of Π v ◦ deter- mined by β  preserves the Π v ◦ -conjugacy class of Π e ◦ . Thus, α E/(E\{i}) fixes the Π E/(E\{i}) -conjugacy class of Π e ◦ , as desired. This completes the proof of assertion (i). Next, we verify assertion (ii). Since v is of type (0, 3), it follows from assertion (i), together with Lemma 3.14, (iv), that one may de- fine α| (Π v ) 2 ∈ Out((Π v ) 2 ). Thus, by applying Lemma 3.13, (iii), to α| (Π v ) 2 ∈ Out((Π v ) 2 ), one verifies easily that the first portion of asser- tion (ii) holds. The final portion of assertion (ii) follows immediately from the descriptions given in the four conditions of Lemma 3.8, (ii), together with the various definitions involved. This completes the proof of assertion (ii).  Theorem 3.16 (Outomorphisms preserving tripods). In the no- tation of the beginning of the present §3, let E ⊆ {1, · · · , n} and T ⊆ Π E an E-tripod of Π n [cf. Definition 3.3, (i)]. Let us write Out F (Π n )[T ] ⊆ Out F (Π n ) for the [closed] subgroup of Out F (Π n ) [cf. [CmbCsp], Definition 1.1, (ii)] consisting of F-admissible outomorphisms α of Π n such that the outomorphism of Π E determined by α preserves the Π E -conjugacy class of T ⊆ Π E . Then the following hold: (i) It holds that C Π E (T ) = T × Z Π E (T ). Thus, by applying Lemma 3.10, (i), to outomorphisms of Π E determined by elements of Out F (Π n )[T ], one obtains a natural homomorphism T T : Out F (Π n )[T ] −→ Out(T ). 80 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Let us write Out F (Π n )[T : {C}], Out F (Π n )[T : {|C|}], Out F (Π n )[T : {Δ}], Out F (Π n )[T : {+}] ⊆ Out F (Π n )[T ] for the [closed] subgroups of Out F (Π n )[T ] obtained by forming the respective inverse images via T T of the closed subgroups Out C (T ), Out C (T ) cusp , Out(T ) Δ , Out(T ) + ⊆ Out(T ) [cf. Def- inition 3.4, (i)]. For each subset S ⊆ {C, |C|, Δ, +}, let us write  def Out F (Π n )[T : S] = Out F (Π n )[T : {}] ⊆ Out F (Π n )[T ] ; ∈S def Out FC (Π n )[T : S] = Out F (Π n )[T : S] ∩ Out FC (Π n ) ⊆ Out FC (Π n ) [cf. [CmbCsp], Definition 1.1, (ii)]. Suppose, moreover, that we are given an element σ ∈ S n ⊆ Out(Π n ) [cf. the discussion at the beginning of the present §3] and a lifting σ  ∈ Aut(Π n ) of σ ∈ S n ⊆ Out(Π n ). Write T σ  ⊆ Π σ(E) ∼ for the image of T ⊆ Π E by the isomorphism Π E → Π σ(E) determined by σ  ∈ Aut(Π n ) [which thus implies that T σ  ⊆ Π σ(E) is a σ(E)-tripod of Π n — cf. Remark 3.7.1] and  ] = Out F (Π n )[T ] ∩ Out F (Π n )[T σ  ] ⊆ Out F (Π n ), Out F (Π n )[T, σ def def Out FC (Π n )[T, σ  ] = Out F (Π n )[T, σ  ]∩Out FC (Π n ) ⊆ Out FC (Π n ). ∼ Then the resulting isomorphism T → T σ  is geometric [cf. Definition 3.4, (ii)]. Moreover, we have a commutative dia- gram Out F (Π n )[T, σ  ] ⏐ ⏐ T T  Out F (Π n )[T, σ  ] ⏐ ⏐ T  T σ  ∼ Out(T ) −−−→ Out(T σ  ) — where the upper horizontal equality is an equality of sub- groups of the group Out F (Π n ), and the lower horizontal arrow is the isomorphism obtained by conjugating by the above geo- ∼  ∈ Aut(Π n )]. metric isomorphism T → T σ  [i.e., induced by σ Finally, the equalities  ] = Out FC (Π n )[T ] = Out FC (Π n )[T σ  ] Out FC (Π n )[T, σ hold; if, moreover, one of the following conditions is satisfied, then the equalities Out F (Π n )[T, σ  ] = Out F (Π n )[T ] = Out F (Π n )[T σ  ] hold: COMBINATORIAL ANABELIAN TOPICS II 81 (i-1) (r, n)  = (0, 2). (i-2) T is E-strict [cf. Definition 3.3, (iii)]. (ii) It holds that Out F (Π n )[T : {C, Δ}] = Out F (Π n )[T : {|C|, Δ}] . (iii) Suppose that T is 1-descendable [cf. Definition 3.3, (iv)]. Then it holds that Out FC (Π n )[T : {|C|}] = Out FC (Π n )[T : {|C|, +}] . If, moreover, one of the following conditions is satisfied, then it holds that Out F (Π n )[T : {|C|}] = Out F (Π n )[T : {|C|, +}] : (iii-1) T is 2-descendable [cf. Definition 3.3, (iv)]. (iii-2) There exists a subset E  ⊆ E such that: (iii-2-a) E   = {1, · · · , n};  (iii-2-b) the image p Π E/E  (T ) ⊆ Π E is a cusp-supporting  E -tripod of Π n [cf. Definition 3.3, (i)]. (iv) Let i, j ∈ E be two distinct elements of E; e ∈ Edge(G j∈E\{i},x ) [cf. Definition 3.1, (iii)]; α ∈ Out F (Π n ). Suppose that T arises from e ∈ Edge(G j∈E\{i},x ) [cf. Definition 3.7, (i)], and that the outomorphism of Π E\{i} determined by α preserves the Π E\{i} -conjugacy class of an edge-like subgroup of Π E\{i} associated to e ∈ Edge(G j∈E\{i},x ) [cf. Definition 3.1, (iv)]. Suppose, moreover, that one of the following conditions is sat- isfied: (iv-1) α ∈ Out FC (Π n ). (iv-2) #E ≤ n − 1. (iv-3) e ∈ Cusp(G j∈E\{i},x ). Then α ∈ Out F (Π n )[T ]. Suppose, further, that either condition (iv-1) or condition (iv-2) is satisfied. Then α ∈ Out F (Π n )[T : {C}]; if, in addition, condition (iv-3) is satisfied, then α ∈ Out F (Π n )[T : {|C|}]. (v) Suppose that T is central [cf. Definition 3.7, (ii)]. If n ≥ 4 [i.e., T is 1-descendable], then it holds that Out F (Π n ) = Out FC (Π n )[T : {|C|, Δ, +}] . If n = 3 [i.e., T is not 1-descendable], then it holds that Out FC (Π n ) = Out FC (Π n )[T : {|C|, Δ}] ⊆ Out F (Π n ) = Out F (Π n )[T : {Δ}] ; 82 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI if, moreover, r  = 0, then Out F (Π n ) = Out FC (Π n )[T : {|C|, Δ, +}] . Proof. We begin the proof of Theorem 3.16 with the following claim: Claim 3.16.A: Let E  ⊆ E be a subset such that the   image T E  of T via p Π E/E  : Π E  Π E is an E -tripod. Thus, one verifies easily that one obtains a(n) [neces- ∼ sarily geometric] outer isomorphism T → T E  [induced F by p Π E/E  ]. Then we have an inclusion Out (Π n )[T ] ⊆ Out F (Π n )[T E  ], and, moreover, the diagram Out F (Π n )[T ] ⊆ T T ↓ Out(T ) Out F (Π n )[T E  ] ↓ T T E  ∼ −→ Out(T E  ) — where the lower horizontal arrow is the isomorphism ∼ determined by the isomorphism T → T E  induced by p Π E/E  — commutes. Indeed, this follows immediately from the various definitions involved. This completes the proof of Claim 3.16.A. Next, we verify assertion (i). The equality C Π E (T ) = T × Z Π E (T ) of the first display in assertion (i) follows from Lemma 3.12, (i). Moreover, ∼ the geometricity of the isomorphism T → T σ  follows immediately from the various definitions involved. Next, let us observe that if (r, n)  = (0, 2), then the commutativity of the displayed diagram in assertion (i) and the equalities  ] = Out F (Π n )[T ] = Out F (Π n )[T σ  ] Out F (Π n )[T, σ in assertion (i) may be easily derived from the fact that the closed subgroup Out F (Π n ) ⊆ Out(Π n ) centralizes the closed subgroup S n ⊆ Out F (Π n ) [cf. Theorem 2.3, (iv)]. Moreover, the equalities Out FC (Π n )[T, σ  ] = Out FC (Π n )[T ] = Out FC (Π n )[T σ  ] in assertion (i) may be easily derived from the fact that the closed subgroup Out FC (Π n ) ⊆ Out(Π n ) centralizes the closed subgroup S n ⊆ Out F (Π n ) [cf. [NodNon], Theorem B]. Next, let us observe that if T is E  -strict for some subset E  ⊆ E of cardinality one, then the commutativity of the displayed diagram in assertion (i) follows immediately from Claim 3.16.A and [CbTpI], The- orem A, (i). Thus, it follows from Lemma 3.8, (ii), that, to complete the verification of assertion (i), it suffices to verify, under the assumption that σ  = id, COMBINATORIAL ANABELIAN TOPICS II 83 (a) the commutativity of the displayed diagram in assertion (i) in the case where (r, n) = (0, 2), and T is {1, 2}-strict, and (b) the equalities Out F (Π n )[T, σ  ] = Out F (Π n )[T ] = Out F (Π n )[T σ  ] in assertion (i) in the case where (r, n) = (0, 2), and T is {1, 2}- strict. In particular, to verify assertion (i), we may assume without loss of generality [cf. conditions (2 C ) and (2 N ) of Lemma 3.8, (ii)] that we are in the situation of Lemma 3.11 in the case where we take the “n”, “E” of Lemma 3.11 to be 2, {1, 2}, respectively. Moreover, it follows immediately from Lemma 3.8, (ii), that the Π n -conjugacy classes of T , T σ  coincide with the Π n -conjugacy classes of the closed subgroups new , Π v new of Π n that appear in the statement of Lemma 3.11, re- Π v 2/1 1\2 spectively. Then the above equalities in (b) follows immediately from Lemma 3.11, (x). Moreover, it follows from Lemma 3.11, (viii), (ix), that the composites T → C Π n (T )  C Π n (T )/Z(C Π n (T )), T σ  → C Π n (T σ  )  C Π n (T σ  )/Z(C Π n (T σ  )) are isomorphisms. Thus, the commutativity in (a) follows immediately from Lemma 3.11, (x). This completes the proof of assertion (i). As- sertion (ii) follows from Lemma 3.5. Next, we verify assertion (iii). First, to verify the first displayed equality of assertion (iii), let us observe that since T is 1-descendable, there exists a subset E  ⊆ E such that the image of T ⊆ Π E via    p Π E/E  : Π E  Π E is an E -tripod, and, moreover, #E ≤ n − 1. Thus, it follows immediately from Claim 3.16.A, together with Remark 3.4.1  — by replacing T , E, by p Π E/E  (T ), E , respectively — that, to verify the first displayed equality of assertion (iii), we may assume without loss of generality that E  = {1, · · · , n}. Then the first displayed equality of assertion (iii) follows immediately from Lemma 3.14, (iv); the portion of Lemma 3.15, (i) [where we observe that the “T ” of Lemma 3.15 differs from the T of the present discussion!], concerning “(Π v ) 2 ” [cf. condition (a) of Lemma 3.15]. This completes the proof of the first displayed equality of assertion (iii). Next, suppose that condition (iii-1) is satisfied; thus, there exists a  subset E  ⊆ E such that the image p Π E/E  (T ) ⊆ Π E is an E -tripod, and,  moreover, #E  ≤ n − 2. Then — by replacing T , E by p Π E/E  (T ), E , respectively [and applying Claim 3.16.A] — we may assume without loss of generality that #E ≤ n − 2. Thus, by applying [CbTpI], Theo- rem A, (ii), we conclude that the second displayed equality of assertion (iii) follows immediately from the first displayed equality of assertion (iii). 84 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Next, suppose that condition (iii-2) is satisfied. Then — by re-  placing T , E by the p Π E/E  (T ), E in condition (iii-2) [and applying Claim 3.16.A] — we may assume without loss of generality that E  = {1, · · · , n}, and, moreover, that T is a cusp-supporting E-tripod. Then it follows immediately from Lemma 3.14, (iv); the portion of Lemma 3.15, (i), concerning (Π v ) 2 [cf. condition (b) of Lemma 3.15], that the second displayed equality of assertion (iii) holds. This completes the proof of assertion (iii). Next, we verify assertion (iv). If either condition (iv-1) or condi- tion (iv-3) is satisfied, then one reduces immediately to the case where n = 2, in which case it follows immediately from Lemma 3.13, (i), that α ∈ Out F (Π n )[T ]. If condition (iv-1) is satisfied, then one reduces im- mediately to the case where n = 2, in which case it follows immediately from Lemma 3.13, (i), that α ∈ Out F (Π n )[T : {C}]. If both condition (iv-1) and condition (iv-3) are satisfied, then — by applying a suit- able specialization isomorphism [cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] — one reduces imme- diately to the case where n = 2 and Node(G) = ∅, in which case it fol- lows immediately from Lemma 3.15, (i), that α ∈ Out F (Π n )[T : {|C|}]. Finally, if condition (iv-2) is satisfied, then, by applying [CbTpI], The- orem A, (ii), one reduces immediately to the case where “n” is taken to be n − 1, and condition (iv-1) is satisfied. This completes the proof of assertion (iv). Finally, we verify assertion (v). First, we claim that the following assertion holds: Claim 3.16.B: Out F (Π n ) = Out F (Π n )[T ]. Indeed, to verify Claim 3.16.B, by reordering the factors of X n , we may assume without loss of generality that E = {1, 2, 3}. Let α  ∈ F Aut (Π n ). Then since n ≥ 3, it follows immediately from [CbTpI], Theorem A, (ii), together with Lemma 3.2, (iv), that the outomorphism  preserves the Π 2/1 -conjugacy class of cuspidal of Π 2/1 determined by α subgroups of Π 2/1 associated to the [unique — cf. Lemma 3.2, (ii)] diagonal cusp. Thus, it follows immediately from assertion (iv) in the case where condition (iv-3) is satisfied that the outomorphism of Π 3 determined by α  preserves the Π 3 -conjugacy class of T ⊆ Π 3 . This completes the proof of Claim 3.16.B. Next, we claim that the following assertion holds: Claim 3.16.C: Out F (Π n )[T ] = Out F (Π n )[T : {Δ}]. Indeed, since n ≥ 3, this follows immediately from Theorem 2.3, (iv), together with a similar argument to the argument used in the proof of [CmbCsp], Corollary 3.4, (i). This completes the proof of Claim 3.16.C. Now it follows immediately from Claims 3.16.B, 3.16.C that we have an equality Out F (Π n ) = Out F (Π n )[T : {Δ}]. Thus, it follows from assertion (ii) and the first displayed equality of assertion (iii), together COMBINATORIAL ANABELIAN TOPICS II 85 with Theorem 2.3, (ii), that, to complete the proof of the content of the first two displays of assertion (v), it suffices to verify the equality Out FC (Π n ) = Out FC (Π n )[T : {C}]. On the other hand, this follows im- mediately from the portion of Lemma 3.15, (i), concerning α| T . [Note that one verifies easily that every central tripod arises from a cusp.] Thus, it remains to verify the equality of the final display of assertion (v). In light of what has already been verified [cf. also Theorem 2.3, (ii)], to verify the final equality of assertion (v), it suffices to verify the condition “+” on the right-hand side of this equality. On the other hand, it follows immediately — by replacing an element of the left-hand side of the equality under consideration by a composite of the element with a suitable outomorphism arising from an element of Out FC (Π 4 ) [cf. the equality of the first display of assertion (v)] — from [CmbCsp], Lemma 2.4, that it suffices to verify the condition “+” on an element of the left-hand side of the equality under consideration that induces the identity automorphism on Cusp(G). Then the equality under consideration follows immediately, in light of the assumption that r  = 0, by first applying Lemma 3.15, (i) [in the case where we take the “E” of loc. cit. to be a subset of E of cardinality two, and we apply the argument involving specialization isomorphisms applied in the proof of assertion (iv)], and then applying Lemma 3.15, (i), (ii) [in the case where we take the “E” of loc. cit. to be E]. This completes the proof of assertion (v).  Remark 3.16.1. Theorem 3.16, (i), may be regarded as a general- ization of [CmbCsp], Corollary 1.10, (ii). On the other hand, Theo- rem 3.16, (v), may be regarded as a more precise version of [CmbCsp], Corollary 3.4. Theorem 3.17 (Synchronization of tripods in two dimensions). In the notation of Theorem 3.16, suppose that n = 2, and that #E = 1; thus, one may regard the E-tripod T of Π n as a verticial subgroup ∼ of Π E → Π G associated to a vertex v T ∈ Vert(G) of type (0, 3) [cf. Definition 3.1, (ii)]. Let E  ⊆ {1, · · · , n} and T  ⊆ Π E  an E  -tripod of Π n . Then the following hold: (i) Suppose that there exists an edge e ∈ E(v T ) from which T  arises [cf. Definition 3.7, (i)]. [Thus, it holds that E  = {1, 2}.] Then it holds that Out FC (Π n )[T : {|C|, Δ}] ⊆ Out FC (Π n )[T  : {|C|, Δ, +}] [cf. the notational conventions of Theorem 3.16, (i)]. More- over, there exists a geometric [cf. Definition 3.4, (ii)] outer 86 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ isomorphism T → T  such that the diagram Out FC (Π n )[T : {|C|, Δ}] Out FC (Π n )[T  : {|C|, Δ, +}] ⊆ T T ↓ ↓ T T  ∼ −→ Out(T ) Out(T  ) [cf. the notation of Theorem 3.16, (i)] — where the lower horizontal arrow is the isomorphism induced by the outer iso- ∼ morphism in question T → T  — commutes. (ii) Suppose that #E  = 1. Thus, one may regard the E  -tripod ∼ T  of Π n as a verticial subgroup of Π E  → Π G associated to a vertex v T  ∈ Vert(G) of type (0, 3). Suppose, moreover, that N (v T ) ∩ N (v T  )  = ∅. Then there exists a geometric [cf. Def- ∼ inition 3.4, (ii)] outer isomorphism T → T  such that if we write Out FC (Π n )[T, T  : {|C|, Δ}] = Out FC (Π n )[T : {|C|, Δ}] ∩ Out FC (Π n )[T  : {|C|, Δ}] , def then the diagram Out FC (Π n )[T, T  : {|C|, Δ}] ⏐ ⏐ T T  Out(T ) Out FC (Π n )[T, T  : {|C|, Δ}] ⏐ ⏐ T   T ∼ −−−→ Out(T  ) — where the lower horizontal arrow is the isomorphism induced ∼ by the outer isomorphism in question T → T  — commutes. Proof. First, we verify assertion (i). Let us observe that the inclu- sion Out FC (Π n )[T : {|C|}] ⊆ Out FC (Π n )[T  ], hence also the inclusion Out FC (Π n )[T : {|C|, Δ}] ⊆ Out FC (Π n )[T  ], follows immediately from Theorem 3.16, (iv), in the case where condition (iv-1) is satisfied. Thus, one verifies easily from Lemma 3.15, (i), (ii) [cf. also Lemma 3.14, (iv)], that the remainder of assertion (i) holds. This completes the proof of assertion (i). Next, we verify assertion (ii). It follows immediately from [CmbCsp], Proposition 1.2, (iii), that we may assume without loss of generality that E  = E. Write T  ⊆ Π n for the {1, 2}-tripod of Π n arising from e ∈ N (v T ) ∩ N (v T  ). Then it follows from assertion ∼ ∼ (i) that there exist geometric outer isomorphisms T → T  , T  → T  that satisfy the condition of assertion (i) [i.e., for the pairs (T, T  ) and (T  , T  )]. Thus, one verifies easily that the [necessarily geometric] outer ∼ ∼ isomorphism T → T  ← T  obtained by forming the composite of these two outer isomorphisms satisfies the condition of assertion (ii). This completes the proof of assertion (ii).  COMBINATORIAL ANABELIAN TOPICS II 87 Theorem 3.18 (Synchronization of tripods in three or more dimensions). In the notation of Theorem 3.16, suppose that n ≥ 3. Then the following hold: (i) It holds that Out FC (Π n )[T : {|C|}] = Out FC (Π n )[T : {|C|, Δ}] [cf. the notational conventions of Theorem 3.16, (i)]. If, more- over, n ≥ 4 or r  = 0, then it holds that Out FC (Π n )[T : {|C|}] = Out FC (Π n )[T : {|C|, Δ, +}] [cf. the notational conventions of Theorem 3.16, (i)]. (ii) Let E  ⊆ {1, · · · , n} and T  ⊆ Π E  an E  -tripod of Π n . Then there exists a geometric [cf. Definition 3.4, (ii)] outer iso- ∼ morphism T → T  such that if we write Out FC (Π n )[T, T  : {|C|}] = Out FC (Π n )[T : {|C|}] ∩ Out FC (Π n )[T  : {|C|}] , then the diagram def Out FC (Π n )[T, T  : {|C|}] ⏐ ⏐ T T  Out(T ) Out FC (Π n )[T, T  : {|C|}] ⏐ ⏐ T   T ∼ −−−→ Out(T  ) [cf. the notation of Theorem 3.16, (i)] — where the lower horizontal arrow is the isomorphism induced by the outer iso- ∼ morphism in question T → T  — commutes. Proof. First, we verify the first displayed equality of assertion (i). Ob- serve that it follows immediately from Lemma 3.8, (i), together with a similar argument to the argument applied in the proof of the first dis- played equality of Theorem 3.16, (iii), that we may assume without loss of generality that T is E-strict, which thus implies that #E ∈ {1, 2, 3} [cf. Lemma 3.8, (ii)]. Now we apply induction on 3 − #E ∈ {0, 1, 2}. If 3 − #E = 0, i.e., T is central [cf. Lemma 3.8, (ii)], then the first dis- played equality of assertion (i) follows immediately from Theorem 3.16, (v). Now suppose that 3 − #E > 0, and that the induction hypoth- esis is in force. Let α ∈ Out FC (Π n )[T : {|C|}]. Then it follows im- mediately from Lemma 3.15, (i), (ii) [cf. also conditions (1), (2) of Lemma 3.15, (ii), where we note that the E, E  , T , T  of the present discussion correspond, respectively, to the “E \ {i}”, “E”, “Π v ”, “T ” of Lemma 3.15], that there exist a subset E ⊆ E  ⊆ {1, · · · , n} and an E  -tripod T  ⊆ Π E  such that 3 − #E  < 3 − #E, T  ⊆ Π E  is E  -strict, and α ∈ Out FC (Π n )[T  : {|C|}] [cf. Lemma 3.15, (i)]. Thus, it follows immediately from the induction hypothesis that α ∈ Out FC (Π n )[T  : {|C|, Δ}]. In particular, it follows immediately from Lemma 3.15, (ii), 88 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI that — for a suitable choice of the pair (E  , T  ) [cf. the statement of Lemma 3.15, (ii)] — the actions of α on T and T  may be related by means of a geometric outer isomorphism, which thus implies that α ∈ Out FC (Π n )[T : {|C|, Δ}] [cf. Remark 3.4.1]. This completes the proof of the first displayed equality of assertion (i). Next, we verify assertion (ii). First, we claim that the following assertion holds: Claim 3.18.A: If both T and T  are central, then the pair (T, T  ) satisfies the property stated in assertion (ii). Indeed, this assertion follows immediately from the commutativity of the displayed diagram of Theorem 3.16, (i). Next, we claim that the following assertion holds: Claim 3.18.B: Suppose that T is E-strict, and that #E  = 3 [i.e., #E ∈ {1, 2} — cf. Lemma 3.8, (ii)]. Then there exist a subset E  E  ⊆ {1, · · · , n} and an E  -tripod T  ⊆ Π E  such that T  is E  -strict, Out FC (Π n )[T : {|C|}] ⊆ Out FC (Π n )[T  : {|C|}], and, moreover, the pair (T, T  ) satisfies the property stated in assertion (ii) [i.e., where one takes “T  ” to be T  ]. Indeed, this follows immediately from Lemma 3.15, (i), (ii) [cf. also conditions (1), (2) of Lemma 3.15, (ii), where we note that the E, E  , T , T  of the present discussion correspond, respectively, to the “E \ {i}”, “E”, “Π v ”, “T ” of Lemma 3.15], together with the first displayed equality of assertion (i). This completes the proof of Claim 3.18.B. To verify assertion (ii), let us observe that it follows immediately from Lemma 3.8, (i), together with a similar argument to the argument applied in the proof of the first displayed equality of Theorem 3.16, (iii), that we may assume without loss of generality that T is E-strict; in par- ticular, #E ∈ {1, 2, 3} [cf. Lemma 3.8, (ii)]. Next, let us observe that, by comparing two arbitrary tripods of Π n to a fixed central tripod of Π n [and applying Theorem 3.16, (v)], one may reduce immediately to the case where T  is central. Moreover, by successive application of Claim 3.18.B, one reduces immediately to the case where both T and T  are central, which was verified in Claim 3.18.A. This completes the proof of assertion (ii). Finally, the second displayed equality of assertion (i) follows immediately from assertion (ii), together with Theorem 3.16, (v). This completes the proof of Theorem 3.18.  Definition 3.19. Suppose that n ≥ 3. Let us write Π tpd COMBINATORIAL ANABELIAN TOPICS II 89 for the i-central E-tripod of Π n [cf. Definitions 3.3, (i); 3.7, (ii)], where E ⊆ {1, . . . , n} is a subset of cardinality 3, and i ∈ E. Then it follows from Theorem 3.16, (i), (v), that one has a natural homomorphism T Π tpd : Out FC (Π n ) = Out FC (Π n )[Π tpd : {|C|, Δ}] −→ Out C (Π tpd ) Δ [cf. Definition 3.4, (i)], which is in fact independent of E and i [cf. Theorem 3.16, (i)]. We shall refer to this homomorphism as the tripod homomorphism associated to Π n and write Out FC (Π n ) geo ⊆ Out FC (Π n ) for the kernel of this homomorphism [cf. Remark 3.19.1 below]. Note that it follows from Theorem 3.16, (v), that if n ≥ 4 or r  = 0, then the image of the tripod homomorphism is contained in Out C (Π tpd ) Δ+ ⊆ Out C (Π tpd ) Δ [cf. Definition 3.4, (i)]. If n ≥ 4 or r  = 0, then T Π tpd may also be regarded as a homomorphism defined on Out F (Π n ) (= Out FC (Π n ) — cf. Theorem 2.3, (ii)); in this case, we shall write def Out F (Π n ) geo = Out FC (Π n ) geo . Remark 3.19.1. Let us recall that if we write π 1 ((M g,[r] ) Q ) for the étale fundamental group of the moduli stack (M g,[r] ) Q of hyperbolic curves of type (g, r) over Q [cf. the discussion entitled “Curves” in “Notations and Conventions”], then we have a natural outer homo- morphism π 1 ((M g,[r] ) Q ) −→ Out FC (Π n ) . Suppose that n ≥ 4. Then Out FC (Π n ) = Out F (Π n ) does not de- pend on n [cf. Theorem 2.3, (ii); [NodNon], Theorem B]. Morever, one verifies easily that the image of the geometric fundamental group π 1 ((M g,[r] ) Q ) ⊆ π 1 ((M g,[r] ) Q ) — where we use the notation Q to denote an algebraic closure of Q — via the above displayed outer homomor- phism is contained in the kernel Out FC (Π n ) geo ⊆ Out FC (Π n ) of the tripod homomorphism associated to Π n [cf. Definition 3.19]. Thus, the outer homomorphism of the above display fits into a commutative diagram of profinite groups 1 −−−→ π 1 ((M g,[r] ) Q ) −−−→ π 1 ((M g,[r] ) Q ) −−−→ ⏐ ⏐ ⏐ ⏐   1 −−−→ Out F (Π n ) geo −−−→ Out F (Π n ) Gal(Q/Q) ⏐ ⏐  −−−→ 1 T tpd Π −− −→ Out C (Π tpd ) Δ+ — where the horizontal sequences are exact. In §4 below, we shall ver- ify that the lower right-hand horizontal arrow T Π tpd is surjective [cf. Corollary 4.15 below]. On the other hand, if Σ is the set of all prime numbers, then it follows from Belyi’s Theorem that the right-hand vertical arrow is injective; moreover, the surjectivity of the right-hand 90 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI vertical arrow has been conjectured in the theory of the Grothendieck- Teichmüller group. From this point of view, one may regard the quo- T Πtpd tient Out F (Π n )  Out C (Π tpd ) Δ+ as a sort of arithmetic quotient of Out F (Π n ) and the subgroup Out F (Π n ) geo ⊆ Out F (Π n ) as a sort of geometric portion of Out F (Π n ). Definition 3.20. Let m be a positive integer and Y log a stable log curve over (Spec k) log . For each nonnegative integer i, write Y Π i for the “Π i ” that occurs in the case where we take “X log ” to be Y log . Then we shall say that an isomorphism (respectively, outer isomorphism) ∼ Π 1 → Y Π 1 is m-cuspidalizable if it arises from a [necessarily unique, up to a permutation of the m factors, by [NodNon], Theorem B] PFC- ∼ admissible [cf. [CbTpI], Definition 1.4, (iii)] isomorphism Π m → Y Π m . Proposition 3.21 (Tripod homomorphisms and finite étale cov- erings). Let Y log be a stable log curve over (Spec k) log and Y log → X log a finite log étale covering over (Spec k) log . For each positive inte- ger i, write Y i log (respectively, Y Π i ) for the “X i log ” (respectively, “Π i ”) that occurs in the case where we take “X log ” to be Y log . Suppose that Y log → X log is geometrically pro-Σ and geometrically Galois, i.e., Y log → X log determines an injection Y Π 1 → Π 1 [that is well- defined up to Π 1 -conjugation] whose image is normal. Let α  be an Y automorphism of Π 1 that preserves Π 1 ⊆ Π 1 . Suppose, moreover, that  is n-cuspidalizable [cf. the outomorphism α of Π 1 determined by α Definition 3.20]. Then the following hold: (i) The outomorphism Y α of Y Π 1 determined by α  is n-cuspidali- zable [cf. Definition 3.20]. (ii) Suppose that n ≥ 3. Let Π tpd ⊆ Π 3 , Y Π tpd ⊆ Y Π 3 be 1-central [{1, 2, 3}-]tripods [cf. Definitions 3.3, (i); 3.7, (ii)] of Π n , Y Π n , respectively. Write α n , Y α n for the respective FC-admissible outomorphisms of Π n , Y Π n determined by the n-cuspidalizable outomorphisms α, Y α [cf. (i)]. Then there exists a geometric ∼ [cf. Definition 3.4, (ii)] outer isomorphism φ tpd : Π tpd → Y Π tpd such that the outomorphism T Π tpd (α n ) [cf. Definition 3.19] of Π tpd is compatible with the outomorphism T Y Π tpd ( Y α n ) [cf. Definition 3.19] of Y Π tpd relative to φ tpd . Proof. First, let us observe that, to verify Proposition 3.21 — by apply- ing a suitable specialization isomorphism [cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] — we may assume without loss of generality that X log and Y log are smooth log curves over (Spec k) log . Write (U X ) n , (U Y ) n for the [open subschemes COMBINATORIAL ANABELIAN TOPICS II 91 of X n , Y n determined by the] 1-interiors [cf. [MzTa], Definition 5.1, (i)] of X n log , Y n log , respectively. [Here, we note that in the present situ- ation, the 0-interior of (Spec k) log , hence also of X n log , Y n log , is empty!] def def Thus, one verifies easily that U X = (U X ) 1 , U Y = (U Y ) 1 are hyper- bolic curves over k, and that (U X ) n , (U Y ) n are naturally isomorphic to the n-th configuration spaces of U X , U Y , respectively. Write U X ×n , U Y ×n for the respective fiber products of n copies of U X , U Y over k; Π ×n 1 , Y ×n Y Π 1 for the respective direct products of n copies of Π 1 , Π 1 ; V n for the fiber product of the natural open immersion (U X ) n → U X ×n and the natural finite étale covering U Y ×n → U X ×n . Then one verifies easily that the resulting open immersion V n → U Y ×n factors through the nat- ural open immersion (U Y ) n → U Y ×n , i.e., we obtain an open immersion V n → (U Y ) n . That is to say, whereas (U Y ) n is the open subscheme of U Y ×n obtained by removing the various diagonals of U Y ×n , the scheme V n may be thought of as the open subscheme of U Y ×n obtained by removing the various Galois conjugates of these diagonals, relative to the action of the Galois group Gal(U Y ×n /U X ×n ) = Gal(U Y /U X ) ×n . In particular, we obtain a natural outer isomorphism and outer surjection ∼ Y ×n Π 1 ← Π V n  Y Π n Π n × Π ×n 1 — where we write Π V n for the maximal pro-Σ quotient of the étale fundamental group of V n . Now we verify assertion (i). Let α  n be an FC-admissible automor-  of Π 1 with respect phism of Π n that lies over the automorphism α  n is to each of the n natural projections Π n  Π 1 . Then since α FC-admissible and commutes with the image of the natural inclusion S n → Out(Π n ) [cf. [NodNon], Theorem B], one verifies easily, in light of the description given above of V n , that the outomorphism of Y ×n Π n × Π ×n Π 1 induced by α  n and Y α preserves the inertia subgroups 1 associated to each irreducible component of the complement U Y ×n \ V n . Thus, since [by the Zariski-Nagata purity theorem] the inertia sub- groups of the irreducible components of the complement (U Y ) n \ V n normally topologically generate the kernel of the above outer surjec- tion Π V n  Y Π n , we conclude, by applying the morphisms of the above Y ×n display, that the outomorphism of Π n × Π ×n Π 1 induced by α  n and 1 Y Y α determines an FC-admissible outomorphism of Π n . Moreover, one verifies easily that the resulting outomorphism of Y Π n lies over the outomorphism Y α of Y Π 1 . This completes the proof of assertion (i). Next, we verify assertion (ii). First, let us observe that the natural inclusion Π tpd → Π 3 , together with the trivial homomorphism Π tpd → ({1} →) Y Π ×3 1 [cf. Definition 3.3, (ii); Lemma 3.6, (v); Definition 3.7, Y ×3 ∼ (ii)], determines an injection Π tpd → Π 3 × Π ×3 Π 1 ← Π V 3 . Moreover, 1 it follows immediately from the fact that the blow-up operation that gives rise to a central tripod is compatible with étale localization [cf. the 92 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI discussion of [CmbCsp], Definition 1.8] that — after possibly replacing Y tpd Π ⊆ Y Π 3 by a suitable Y Π 3 -conjugate of Y Π tpd — the composite of this injection Π tpd → Π V 3 with the natural outer surjection Π V 3  Y Π 3 of the above display determines a geometric outer [cf. Lemma 3.12, ∼ (i)] isomorphism φ tpd : Π tpd → Y Π tpd ⊆ Y Π 3 . On the other hand, one verifies easily [cf. the construction of Y α n given in the proof of assertion (i)] that this outer isomorphism φ tpd satisfies the property stated in assertion (ii). This completes the proof of assertion (ii).  Corollary 3.22 (Non-surjectivity result). In the notation of The- orem 3.16, suppose that (g, r) ∈ {(0, 3); (1, 1)}. Then the natural in- jection Out FC (Π 2 ) → Out FC (Π 1 ) of [NodNon], Theorem B, is not surjective. Proof. First, let us observe — by considering a suitable stable log curve of type (g, r) over (Spec k) log and applying a suitable special- ization isomorphism [cf. the discussion preceding [CmbCsp], Defini- tion 2.1, as well as [CbTpI], Remark 5.6.1] — that, to verify Corol- lary 3.22, we may assume without loss of generality that G is totally degenerate [cf. [CbTpI], Definition 2.3, (iv)], i.e., that every vertex of G is a tripod of X n log [cf. Definition 3.1, (v)]. Note that [since (g, r) ∈ {(0, 3); (1, 1)}] this implies that #Vert(G) ≥ 2. Let us fix def a vertex v 0 ∈ Vert(G) and write α v 0 = id G| v 0 ∈ Aut |grph| (G| v 0 ) [cf. [CbTpI], Definitions 2.1, (iii), and 2.6, (i); Remark 4.1.2 of the present monograph]. For each v ∈ Vert(G) \ {v 0 }, let α v ∈ Aut |grph| (G| v ) be a nontrivial automorphism of G| v such that α v ∈ Out C (Π G| v ) Δ , and, moreover, χ G| v (α v ) = 1 [cf. [CbTpI], Definition 3.8, (ii)]. Here, we note that since the image of the natural outer Galois representation of the absolute Galois group of Q associated to P 1 Q \ {0, 1, ∞} is contained in “Out C (−) Δ ”, by considering a nontrivial element of this image whose image via the cyclotomic character is trivial, one verifies immediately [e.g., by applying [LocAn], Theorem A] that such an automorphism α v ∈ Aut |grph| (G| v ) always exists. Then it follows immediately from [CbTpI], Theorem B, (iii), that there exists an automorphism α ∈ Aut |grph| (G) such that ρ Vert G (α) = (α v ) v∈Vert(G) . Now assume that there exists an outomorphism α 2 ∈ Out FC (Π 2 ) such that α ∈ Aut |grph| (G) ∼ (⊆ Out(Π G ) ← Out(Π 1 )) is equal to the image of α 2 via the injection in question Out FC (Π 2 ) → Out FC (Π 1 ). Then, for each v ∈ Vert(G), since α v ∈ Out C (Π G| v ) Δ , and α ∈ Aut |grph| (G), it follows immediately from the various definitions involved that α 2 ∈ Out FC (Π 2 )[Π v : {|C|, Δ}] — where we use the notation Π v to denote a verticial subgroup of COMBINATORIAL ANABELIAN TOPICS II ∼ 93 def Π G ← Π 1 associated to v ∈ Vert(G). Thus, since α v 0 = id G| v 0 , it fol- lows from Theorem 3.17, (ii), that α v = id G| v for every v ∈ Vert(G), in contradiction to the fact that for v ∈ Vert(G) \ {v 0 } ( = ∅), the auto- morphism α v ∈ Aut |grph| (G| v ) is nontrivial. This completes the proof of Corollary 3.22.  Remark 3.22.1. (i) Let us recall from [NodNon], Corollary 6.6, that, in the dis- crete case, the homomorphism that corresponds to the homo- morphism discussed in Corollary 3.22 is, in fact, surjective; moreover, this surjectivity may be regarded as an immediate consequence of the Dehn-Nielsen-Baer theorem — cf. the proof of [CmbCsp], Theorem 5.1, (ii). This phenomenon illustrates that, in general, analogous constructions in the discrete and profinite cases may in fact exhibit quite different behavior. (ii) In the context of (i), we recall another famous example of sub- stantially different behavior in the discrete and profinite cases: As is well-known, in classical algebraic topology, singular co- homology with coefficients in Z yields a “good” cohomology theory with coefficients in Z. On the other hand, in the 1960’s, Serre gave an argument involving supersingular elliptic curves in characteristic p > 0 which shows that such a “good” coho- mology theory with coefficients in Z [or even in Z p !] cannot exist for smooth varieties of positive characteristic. (iii) In [Lch], various conjectures concerning [in the notation of the present monograph] the profinite group “Out(Π 1 )” were intro- duced. However, at the time of writing, the authors of the present monograph were unable to find any justification for the validity of these conjectures that goes beyond the observa- tion that the discrete analogues of these conjectures are indeed valid. That is to say, there does not appear to exist any justi- fication for excluding the possibility that — just as in the case of the examples discussed in (i), (ii), i.e., the Dehn-Nielsen- Baer theorem and singular cohomology with coefficients in Z — the discrete and profinite cases exhibit substantially differ- ent behavior. In particular, it appears to the authors that it is desirable that this issue be addressed in a satisfactory fashion in the context of these conjectures. Remark 3.22.2. As discussed in Remark 3.22.1, (i), in the discrete case, the homomorphism that corresponds to the homomorphism dis- cussed in Corollary 3.22 is, in fact, bijective. The proof of Corollary 3.22 94 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI fails in the discrete case for the following reason: The pro-Σ “Π 1 ” of a tripod admits nontrivial C-admissible outomorphisms that commute with the outer modular symmetries and, moreover, lie in the kernel of the cyclotomic character [cf. the proof of Corollary 3.22]. By contrast, the discrete “Π 1 ” of a tripod does not admit such outomorphisms. In- deed, it follows from a classical result of Nielsen [cf. [CmbCsp], Remark 5.3.1] that the discrete “Out C (Π 1 ) cusp ” in the case of a tripod is a finite group of order 2 whose unique nontrivial element arises from complex conjugation. Remark 3.22.3. It follows from [NodNon], Theorem B, together with Corollary 3.22, that if (g, r) ∈ {(0, 3); (1, 1)}, then the homomorphism Out FC (Π n+1 ) → Out FC (Π n ) of [NodNon], Theorem B, fits into the following sequences of homomorphisms of profinite groups: If r  = 0, then for any n ≥ 3, ≃ ≃ ? ∼ Out FC (Π n ) → Out FC (Π 3 ) → Out FC (Π 2 ) → Out FC (Π 1 ) . If r = 0, then for any n ≥ 4, ∼ ≃ ? ≃ ≃ ? Out FC (Π n ) → Out FC (Π 4 ) → Out FC (Π 3 ) → Out FC (Π 2 ) → Out FC (Π 1 ) . Definition 3.23. Let Σ 0 be a nonempty set of prime numbers and G 0 a semi-graph of anabelioids of pro-Σ 0 PSC-type. Write Π G 0 for the [pro-Σ 0 ] fundamental group of G 0 . (i) Let H be a semi-graph of anabelioids of pro-Σ 0 PSC-type, ∼ S ⊆ Node(H), and φ : H S → G 0 [cf. [CbTpI], Definition 2.8, for more on this notation] an isomorphism [of semi-graphs of anabelioids of PSC-type]. Then we shall refer to the triple (H, S, φ) as a degeneration structure on G 0 . (ii) Let (H 1 , S 1 , φ 1 ), (H 2 , S 2 , φ 2 ) be two degeneration structures on G 0 [cf. (i)]. Then we shall write (H 2 , S 2 , φ 2 ) (H 1 , S 1 , φ 1 ) if there exist a subset S 2,1 ⊆ S 2 of S 2 and a(n) [uniquely de- termined, by φ 1 and φ 2 ! — cf. [CmbGC], Proposition 1.5, ∼ (ii)] isomorphism φ 2,1 : (H 2 ) S 2,1 → H 1 [i.e., a degeneration structure (H 2 , S 2,1 , φ 2,1 ) on H 1 ] such that φ 2,1 maps S 2 \ S 2,1 bijectively onto S 1 , and the diagram ∼ ((H 2 ) S 2,1 ) S 2 \S 2,1 −−−→ (H 1 ) S 1 ⏐ ⏐ ⏐ ⏐     φ 1 (H 2 ) S 2 φ 2 −−−→ ∼ G 0 COMBINATORIAL ANABELIAN TOPICS II 95 — where the upper horizontal arrow is the isomorphism in- duced by φ 2,1 , and the left-hand vertical arrow is the natural isomorphism — commutes. [Here, we note that the subset S 2,1 is also uniquely determined by φ 1 and φ 2 — cf. [CmbGC], Proposition 1.2, (i).] (iii) Let (H 1 , S 1 , φ 1 ), (H 2 , S 2 , φ 2 ) be two degeneration structures on G 0 [cf. (i)]. Then we shall say that (H 1 , S 1 , φ 1 ) is co-Dehn to (H 2 , S 2 , φ 2 ) if there exists a degeneration structure (H 3 , S 3 , φ 3 ) on G 0 such that (H 3 , S 3 , φ 3 ) (H 1 , S 1 , φ 1 ); (H 3 , S 3 , φ 3 ) (H 2 , S 2 , φ 2 ) [cf. (ii)]. (iv) Let (H, S, φ) be a degeneration structure on G 0 [cf. (i)] and α ∈ Out(Π G 0 ). Then we shall say that α is an (H, S, φ)-Dehn multi-twist of G 0 if α is contained in the image of the composite ∼ ∼ Dehn(H) → Out(Π H ) ← Out(Π H S ) → Out(Π G 0 ) — where the first arrow is the natural inclusion [cf. [CbTpI], Definition 4.4], the second arrow is the isomorphism deter- mined by Φ H S [cf. [CbTpI], Definition 2.10], and the third arrow is the isomorphism determined by φ. We shall say that α is a nondegenerate (respectively, positive definite) (H, S, φ)- Dehn multi-twist of G 0 if α is the image of a nondegenerate [cf. [CbTpI], Definition 5.8, (ii)] (respectively, positive definite [cf. [CbTpI], Definition 5.8, (iii)]) profinite Dehn multi-twist of H via the above composite. (v) Let m be a positive integer and Y log a stable log curve over (Spec k) log . If m ≥ 2, then suppose that Σ 0 is either equal to Primes or of cardinality one. For each nonnegative integer i, write Y Π i (respectively, H) for the “Π i ” (respectively, “G”) that occurs in the case where we take “X log ” to be Y log . Then we shall say that a degeneration structure (H, S, φ) on G [cf. (i)] is m-cuspidalizable if the composite Y ∼ Φ H S ∼ φ ∼ ∼ Π 1 −→ Π H ←− Π H S −→ Π G ←− Π 1 — where the first and fourth arrows are the natural outer iso- morphisms [cf. Definition 3.1, (ii)], and the second arrow Φ H S is the natural outer isomorphism of [CbTpI], Definition 2.10 — is m-cuspidalizable [cf. Definition 3.20]. Remark 3.23.1. One interesting open problem in the theory of profi- nite Dehn multi-twists developed in [CbTpI], §4, is the following: In 96 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI the notation of Definition 3.23, for i = 1, 2, let (H i , S i , φ i ) be a de- generation structure on G 0 [cf. Definition 3.23, (i)]; α i ∈ Out(Π G 0 ) a nondegenerate (H i , S i , φ i )-Dehn multi-twist [cf. Definition 3.23, (iv)]. Then: Suppose that α 1 commutes with α 2 . Then is (H 1 , S 1 , φ 1 ) co-Dehn to (H 2 , S 2 , φ 2 ) [cf. Definition 3.23, (iii)]? It is not clear to the authors at the time of writing whether or not this question may be answered in the affirmative. Nevertheless, we are able to obtain a partial result in this direction [cf. Corollary 3.25 below]. Proposition 3.24 (Compatibility of tripod homomorphisms). Suppose that n ≥ 3. Then the following hold: (i) Let Y log be a stable log curve over (Spec k) log . For each non- negative integer i, write Y Π i (respectively, H) for the “Π i ” (re- spectively, “G”) that occurs in the case where we take “X log ” to be Y log . Let (H, S, φ) be an n-cuspidalizable degeneration ∼ structure on G [cf. Definition 3.23, (i), (v)]; φ n : Y Π n → Π n a PFC-admissible outer isomorphism [cf. [CbTpI], Definition 1.4, (iii)] that lies over the displayed composite isomorphism of Definition 3.23, (v); Π tpd ⊆ Π 3 , Y Π tpd ⊆ Y Π 3 1-central [{1, 2, 3}-]tripods [cf. Definitions 3.3, (i); 3.7, (ii)] of Π n , Y Π n , respectively. Then there exists an outer isomorphism ∼ φ tpd : Y Π tpd → Π tpd such that the diagram ∼ Out FC ( Y Π n ) −−−→ Out FC (Π n ) ⏐ ⏐ ⏐ T ⏐ T Y Πtpd   Πtpd ∼ Out( Y Π tpd ) −−−→ Out(Π tpd ) [cf. Definition 3.19] — where the upper and lower horizontal arrows are the isomorphisms induced by φ n , φ tpd , respectively — commutes, up to inner automorphisms of Out(Π tpd ). In particular, φ n determines an isomorphism ∼ Out FC ( Y Π n ) geo −→ Out FC (Π n ) geo [cf. Definition 3.19]. (ii) If we regard Out FC (Π n ) as a closed subgroup of Out FC (Π 1 ) by means of the natural injection Out FC (Π n ) → Out FC (Π 1 ) of [NodNon], Theorem B, then the closed subgroup Dehn(G) ⊆ ∼ (Aut(G) ⊆) Out(Π G ) ← Out(Π 1 ) [cf. [CbTpI], Definition 4.4] is contained in Out FC (Π n ) geo ⊆ Out FC (Π n ), i.e., Dehn(G) ⊆ Out FC (Π n ) geo . COMBINATORIAL ANABELIAN TOPICS II 97 Proof. First, we verify assertion (i). Let us observe that if the outer isomorphism φ n arises scheme-theoretically as a specialization isomor- phism — cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1 — then the commutativity in question follows immediately from the various definitions involved [cf. also the discus- sion preceding [CmbCsp], Definition 2.1]. Now the general case follows from the observation that the scheme-theoretic case treated above al- lows one to reduce to the case where Y log = X log , and φ n is an FC- admissible outomorphism, in which case the commutativity in question is a tautological consequence of the fact that T Π tpd is a group homomor- phism. This completes the proof of assertion (i). Next, we verify assertion (ii). The inclusion Dehn(G) ⊆ Out FC (Π n ) follows immediately from the fact that every profinite Dehn multi- twist arises scheme-theoretically. Next, we observe that the inclusion Dehn(G) ⊆ Out FC (Π n ) geo may be regarded either as a consequence of the fact that every profinite Dehn multi-twist arises “Q-scheme- theoretically”, i.e., from scheme theory over Q [cf. the commutative diagram of Remark 3.19.1], or as a consequence of the following argu- ment: Observe that it follows immediately from assertion (i), together with [CbTpI], Theorem 4.8, (ii), (iv), that, by applying a suitable spe- cialization isomorphism — cf. the discussion preceding [CmbCsp], Def- inition 2.1, as well as [CbTpI], Remark 5.6.1 — we may assume with- out loss of generality that G is totally degenerate. Then the inclusion Dehn(G) ⊆ Out FC (Π n ) geo follows immediately from Theorem 3.18, (ii) [cf. also Theorem 3.16, (v); [CbTpI], Definition 4.4!]. This completes the proof of assertion (ii).  Corollary 3.25 (Co-Dehn-ness of degeneration structures in the totally degenerate case). In the notation of Theorem 3.16, for i = 1, 2, let Y i log be a stable log curve over (Spec k) log ; H i the “G” that occurs in the case where we take “X log ” to be Y i log ; (H i , S i , φ i ) a 3- cuspidalizable degeneration structure on G [cf. Definition 3.23, (i), (v)]; α i ∈ Out(Π G ) a nondegenerate (H i , S i , φ i )-Dehn multi-twist of G [cf. Definition 3.23, (iv)]. Suppose that α 1 commutes with α 2 , and that H 2 is totally degenerate [cf. [CbTpI], Definition 2.3, (iv)]. Suppose, moreover, that one of the following conditions is satisfied: (a) r  = 0. (b) α 1 and α 2 are positive definite [cf. Definition 3.23, (iv)]. Then (H 1 , S 1 , φ 1 ) is co-Dehn to (H 2 , S 2 , φ 2 ) [cf. Definition 3.23, (iii)], or, equivalently [since H 2 is totally degenerate], (H 2 , S 2 , φ 2 ) (H 1 , S 1 , φ 1 ) [cf. Definition 3.23, (ii)]. 98 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ Proof. For i = 1, 2, write ψ i : Π G → Π H i for the composite outer isomorphism φ i ∼ ψ i : Π G ← Π (H i ) Si Φ (H i ) S i ∼ → Π H i and ψ = ψ 1 ◦ ψ 2 −1 . Write α 1 [H 2 ] ∈ Out(Π H 2 ) for the outomorphism obtained by conjugating α 1 by ψ 2 . First, we claim that the following assertion holds: def Claim 3.25.A: There exists a positive integer a such def that β = α 1 [H 2 ] a ∈ Dehn(H 2 ). Indeed, since α 1 is an (H 1 , S 1 , φ 1 )-Dehn multi-twist of G, the outomor- phism α 1 [H 2 ] of Π H 2 is group-theoretically cuspidal. Thus, since α 1 commutes with α 2 , it follows, in the case of condition (a) (respectively, (b)), from Theorem 1.9, (i) (respectively Theorem 1.9, (ii)), which may be applied in light of [CbTpI], Corollary 5.9, (ii) (respectively, [CbTpI], Corollary 5.9, (iii)), that α 1 [H 2 ] ∈ Aut(H 2 ). In particular, since the underlying semi-graph of H 2 is finite, there exists a positive integer a such that α 1 [H 2 ] a ∈ Aut |grph| (H 2 ) [cf. [CbTpI], Definition 2.6, (i); Remark 4.1.2 of the present monograph]. On the other hand, since α 1 is an (H 1 , S 1 , φ 1 )-Dehn multi-twist of G, it follows immediately from Proposition 3.24, (i), (ii), that the image of α 1 via the tripod homo- morphism associated to Π 3 [cf. Definition 3.19] is trivial. Thus, since H 2 is totally degenerate, and α 1 [H 2 ] a ∈ Aut |grph| (H 2 ), by applying The- orem 3.18, (ii), together with Proposition 3.24, (i), we conclude that β = α 1 [H 2 ] a ∈ Dehn(H 2 ). This completes the proof of Claim 3.25.A. {l} Next, let us fix an element l ∈ Σ. For i ∈ {1, 2}, write H i for the semi-graph of anabelioids of pro-l PSC-type obtained by forming the pro-l completion of H i [cf. [SemiAn], Definition 2.9, (ii)]. Then it fol- lows immediately from Claim 3.25.A, together with [CbTpI], Theorem 4.8, (ii), (iv), that there exists a subset S ⊆ Node(H 2 ) [which may de- {l} pend on l!] such that the automorphism β {l} ∈ Aut(H 2 ) induced by β {l} {l} {l} is contained in Dehn((H 2 ) S ) ⊆ Dehn(H 2 ) ⊆ Aut(H 2 ) [i.e., β {l} is {l} a profinite Dehn multi-twist of (H 2 ) S ], and, moreover, β {l} is nonde- {l} {l} generate as a profinite Dehn multi-twist of (H 2 ) S . Write α 1 for the outomorphism of the pro-l group Π H {l} [which is naturally isomorphic 1 to the maximal pro-l quotient of Π H 1 ] obtained by conjugating α 1 by ∼ ψ 1 and ψ {l} : Π H {l} → Π H {l} for the outer isomorphism induced by ψ 2 1 [cf. the discussion preceding Claim 3.25.A]. Next, we claim that the following assertion holds: Claim 3.25.B: The composite outer isomorphism ψ S : Π (H 2 ) S Φ (H 2 ) S ∼ → ψ ∼ Π H 2 → Π H 1 COMBINATORIAL ANABELIAN TOPICS II 99 ∼ is graphic, i.e., arises from an isomorphism (H 2 ) S → H 1 . ∼ Indeed, let ψ  S : Π (H 2 ) S → Π H 1 be an isomorphism that lifts ψ S . Then it follows immediately from [CmbGC], Proposition 1.5, (ii) — by con- sidering the functorial bijections between the sets “VCN” [cf. [NodNon], Definition 1.1, (iii)] of various connected finite étale coverings of H 1 , (H 2 ) S — that, to verify Claim 3.25.B, it suffices to verify the follow- ing: Let I 2 → (H 2 ) S be a connected finite étale cov- ering of (H 2 ) S that corresponds to a characteristic open subgroup Π I 2 ⊆ Π (H 2 ) S . Write I 1 → H 1 for the connected finite étale covering of H 1 that corre- sponds to the [necessarily characteristic] open sub- def {l} {l} group Π I 1 = ψ  S (Π I 2 ) ⊆ Π H 1 and I 1 , I 2 for the semi-graphs of anabelioids of pro-l PSC-type obtained by forming the pro-l completions of I 1 , I 2 , respec- ∼ tively. Then the outer isomorphism Π I {l} → Π I {l} de- 2 1 termined by ψ  S is graphic. To verify this graphicity, let us first recall that the automorphisms {l} β {l} ∈ Aut((H 2 ) S ) and α 1 ∈ Aut(H 1 ) are nondegenerate profinite Dehn multi-twists. Thus, it follows immediately from Lemma 3.26, (i), (ii), below [cf. also Claim 3.25.A], that there exist liftings β  ∈  1 ∈ Aut(Π H 1 ) of β, α 1 , respectively, and a positive Aut(Π (H 2 ) S ), α integer b such that the outomorphisms γ 2 , γ 1 of Π I {l} , Π I {l} deter- 2 1 {l} mined by β  b , α  b are nondegenerate profinite Dehn multi-twists of I , {l} 1 2 I 1 , respectively, and, moreover, γ 2 and γ 1 a are compatible relative to ∼ the outer isomorphism in question Π I {l} → Π I {l} . Moreover, if condi- 2 1 tion (b) is satisfied, then γ 1 is a positive definite profinite Dehn multi- {l} twist of I 1 [cf. Lemma 3.26, (ii), below]. Thus, it follows, in the case of condition (a) (respectively, (b)), from Theorem 1.9, (i) (respec- tively Theorem 1.9, (ii)), which may be applied in light of [CbTpI], Corollary 5.9, (ii) (respectively, [CbTpI], Corollary 5.9, (iii)), that the ∼ outer isomorphism in question Π I {l} → Π I {l} is graphic. This com- 2 1 pletes the proof of Claim 3.25.B. On the other hand, one verifies eas- ily from the various definitions involved that Claim 3.25.B implies that (H 2 , S 2 , φ 2 ) (H 1 , S 1 , φ 1 ). This completes the proof of Corol- lary 3.25.  Lemma 3.26 (Profinite Dehn multi-twists and pro-Σ comple- tions of finite étale coverings). Let Σ 1 ⊆ Σ 0 be nonempty sets of prime numbers, G 0 a semi-graph of anabelioids of pro-Σ 0 PSC-type, 100 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI H 0 → G 0 a connected finite étale Galois covering that arises from a  ∈ Aut(Π G 0 ). Write normal open subgroup Π H 0 ⊆ Π G 0 of Π G 0 , and α G 1 , H 1 for the semi-graphs of anabelioids of pro-Σ 1 PSC-type obtained by forming the pro-Σ 1 completions of G 0 , H 0 , respectively [cf. [SemiAn], Definition 2.9, (ii)]. Suppose that α  ∈ Aut(Π G 0 ) preserves the normal open subgroup Π H 0 ⊆ Π G 0 corresponding to H 0 → G 0 . Write α G 0 , α H 0 , α G 1 , α H 1 for the respective outomorphisms of Π G 0 , Π H 0 , Π G 1 , Π H 1 in- duced by α  . Suppose, moreover, that α G 0 ∈ Dehn(G 0 ) [cf. [CbTpI], Definition 4.4]. Then the following hold: (i) It holds that α G 1 ∈ Dehn(G 1 ). Moreover, there exists a positive integer a such that a a ∈ Dehn(H 0 ) , α H ∈ Dehn(H 1 ) . α H 0 1 (ii) If, moreover, α G 1 ∈ Dehn(G 1 ) [cf. (i)] is nondegenerate (re- spectively, positive definite) [cf. [CbTpI], Definition 5.8, a (ii), (iii)], then α H ∈ Dehn(H 1 ) [cf. (i)] is nondegenerate 1 (respectively, positive definite). Proof. First, we verify assertion (i). One verifies easily from [NodNon], Lemma 2.6, (i), together with [CbTpI], Corollary 5.9, (i), that there a ∈ Dehn(H 0 ). Now since exists a positive integer a such that α H 0 a α G 0 ∈ Dehn(G 0 ), α H 0 ∈ Dehn(H 0 ), it follows immediately from the a ∈ Dehn(H 1 ). various definitions involved that α G 1 ∈ Dehn(G 1 ), α H 1 This completes the proof of assertion (i). Assertion (ii) follows imme- diately, in the nondegenerate (respectively, positive definite) case, from [NodNon], Lemma 2.6, (i), together with [CbTpI], Corollary 5.9, (ii) (respectively, from Corollary 5.9, (iii), (v)). This completes the proof of Lemma 3.26.  Corollary 3.27 (Commensurator of profinite Dehn multi-twists in the totally degenerate case). In the notation of Theorem 3.16, Definition 3.19 [so n ≥ 3], suppose further that G is totally degener- def ate [cf. [CbTpI], Definition 2.3, (iv)]. Write s : Spec k → (M g,[r] ) k = (M g,[r] ) Spec k [cf. the discussion entitled “Curves” in “Notations and Conventions”] for the underlying (1-)morphism of algebraic stacks of log log def the classifying (1-)morphism (Spec k) log → (M g,[r] ) k = (M g,[r] ) Spec k [cf. the discussion entitled “Curves” in “Notations and Conventions”]  s log for the log scheme of the stable log curve X log over (Spec k) log ; N  s def obtained by equipping N = Spec k with the log structure induced, via log s, by the log structure of (M g,[r] ) k ; N s log for the log stack obtained by  log by the nat- forming the [stack-theoretic] quotient of the log scheme N s ural action of the finite k-group “s × (M g,[r] ) k s”, i.e., the fiber product COMBINATORIAL ANABELIAN TOPICS II 101 over (M g,[r] ) k of two copies of s; N s for the underlying stack of the log stack N s log ; I N s ⊆ π 1 (N s log ) for the closed subgroup of the log fundamen- tal group π 1 (N s log ) of N s log given by the kernel of the natural surjection π 1 (N s log )  π 1 (N s ) [induced by the (1-)morphism N s log → N s obtained (Σ) by forgetting the log structure]; π 1 (N s log ) for the quotient of π 1 (N s log ) by the kernel of the natural surjection from I N s to its maximal pro-Σ Σ . Then the following hold: quotient I N s (i) The natural homomorphism π 1 (N s log ) → Out(Π 1 ) [cf. the natu- ral outer homomorphism of the first display of Remark 3.19.1] (Σ) factors through the quotient π 1 (N s log )  π 1 (N s log ) and the natural inclusion N Out FC (Π n ) geo (Dehn(G)) → Out(Π 1 ) [cf. Propo- sition 3.24, (ii)]. In particular, we obtain a homomorphism (Σ) π 1 (N s log ) −→ N Out FC (Π n ) geo (Dehn(G)) , hence also a homomorphism (Σ) π 1 (N s log ) −→ C Out FC (Π n ) geo (Dehn(G)) . (ii) The second displayed homomorphism of (i) fits into a natural commutative diagram of profinite groups 1 −−−→ Σ I N ⏐ s ⏐  −−−→ (Σ) π 1 (N s log ) ⏐ ⏐  −−−→ π 1 (N s ) −−−→ 1 ⏐ ⏐  1 −−−→ Dehn(G) −−−→ C Out FC (Π n ) geo (Dehn(G)) −−−→ Aut(G) −−−→ 1 [cf. Definition 3.1, (ii), concerning the notation “G”] — where the horizontal sequences are exact, and the vertical arrows are isomorphisms. (iii) Dehn(G) is open in C Out FC (Π n ) geo (Dehn(G)). (iv) We have an equality N Out FC (Π n ) geo (Dehn(G)) = C Out FC (Π n ) geo (Dehn(G)) . Proof. First, we verify assertion (i). The fact that the image of the homomorphism in question is contained in Out FC (Π n ) geo follows imme- diately from the [tautological!] fact that this image arises “Q-scheme- theoretically”, i.e., from scheme theory over Q [cf. the discussion of Remark 3.19.1]. Thus, assertion (i) follows immediately from the fact that the natural homomorphism π 1 (N s log ) → Out(Π 1 ) determines an Σ ∼ isomorphism I N → Dehn(G) [cf. [CbTpI], Proposition 5.6, (ii)]. This s completes the proof of assertion (i). Next, we verify assertion (ii). First, let us observe that it follows from [CbTpI], Theorem 5.14, (iii), that C Out FC (Π n ) geo (Dehn(G)) ⊆ Aut(G). Thus, we obtain a natural homomorphism C Out FC (Π n ) geo (Dehn(G)) → Aut(G), whose kernel contains Dehn(G) [cf. the definition of a profinite 102 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Dehn multi-twist given in [CbTpI], Definition 4.4]. On the other hand, if an element α ∈ C Out FC (Π n ) geo (Dehn(G)) acts trivially on G, then, since G is totally degenerate, it follows immediately from Theorem 3.18, (ii), that α ∈ Dehn(G). This completes the proof of the existence of the lower exact sequence in the diagram of assertion (ii), except for the surjectivity of the third arrow of this sequence. Thus, it follows im- mediately from the proof of assertion (i) that, to complete the proof of assertion (ii), it suffices to verify that the right-hand vertical arrow log π 1 (N s ) → Aut(G) of the diagram is an isomorphism. Write X N  s for the log  whose classifying (1-)morphism is given by stable log curve over N s  log → (M log ) k and Aut  log (X log ) for the the natural (1-)morphism N s g,[r]  s N s N log log log  group of automorphisms of X N  over N s . Then since X , hence also s log X N , is totally degenerate, one verifies easily that the natural homo-  s log morphism Aut N  s log (X N  s ) → Aut(G) is an isomorphism. Thus, it follows immediately from the various definitions involved that the right-hand vertical arrow π 1 (N s ) → Aut(G) of the diagram is an isomorphism. This completes the proof of assertion (ii). Assertion (iii) follows immediately from the exactness of the lower sequence of the diagram of assertion (ii), together with the finiteness of G. Assertion (iv) follows immediately from the fact that the middle vertical arrow of the diagram of assertion (ii) is an isomorphism which factors through N Out FC (Π n ) geo (Dehn(G)) ⊆ C Out FC (Π n ) geo (Dehn(G)) [cf. assertion (i)]. This completes the proof of Corollary 3.27.  Remark 3.27.1. One interesting consequence of Corollary 3.27 is the following: The profinite group Out FC (Π n ) geo [which, as discussed in Remark 3.19.1, may be regarded as the geometric portion of the group of FC-admissible outomorphisms of the configuration space group Π n ], hence also the commensurator C Out FC (Π n ) geo (Dehn(G)), is defined in a purely combinatorial/group-theoretic fashion. In particular, it follows from the commutative diagram of Corollary 3.27, (ii), that this com- mensurator C Out FC (Π n ) geo (Dehn(G)) yields a purely combinatorial/group- theoretic algorithm for reconstructing the profinite groups of scheme- theoretic origin that appear in the upper sequence of this diagram. COMBINATORIAL ANABELIAN TOPICS II 103 4. Glueability of combinatorial cuspidalizations In the present §4, we discuss the glueability of combinatorial cuspidal- izations. The resulting theory may be regarded as a higher-dimensional analogue of the displayed exact sequence of [CbTpI], Theorem B, (iii) [cf. Theorem 4.14, (iii), below, of the present monograph]. This theory implies a certain key surjectivity property of the tripod homomorphism [cf. Corollary 4.15 below]. Finally, we apply this result to construct cuspidalizations of the log fundamental group of a stable log curve over a finite field [cf. Corollary 4.16 below] and to compute certain com- mensurators of the corresponding Galois image in the totally degenerate case [cf. Corollary 4.17 below]. In the present §4, we maintain the notation of the preceding §3 [cf. also Definition 3.1]. In addition, let Σ 0 be a nonempty set of prime numbers and G 0 a semi-graph of anabelioids of pro-Σ 0 PSC-type. Write G 0 for the underlying semi-graph of G 0 and Π G 0 for the [pro-Σ 0 ] funda- mental group of G 0 . Definition 4.1. (i) We shall write Aut |Brch(G 0 )| (G 0 ) ⊆ (Aut |Vert(G 0 )| (G 0 ) ∩ Aut |Node(G 0 )| (G 0 ) ⊆) Aut(G 0 ) [cf. [CbTpI], Definition 2.6, (i)] for the [closed] subgroup of Aut(G 0 ) consisting of automorphisms α of G 0 that induce the identity automorphism of Vert(G 0 ), Node(G 0 ) and, moreover, fix each of the branches of every node of G 0 . Thus, we have a natural exact sequence of profinite groups 1 −→ Aut |grph| (G 0 ) −→ Aut |Brch(G 0 )| (G 0 ) −→ Aut(Cusp(G 0 )) [cf. [CbTpI], Definition 2.6, (i); Remark 4.1.2 of the present monograph]. (ii) Let v ∈ Vert(G 0 ). Then we shall write E(G 0 | v : G 0 ) ⊆ Edge(G 0 | v ) (= Cusp(G 0 | v )) [cf. [CbTpI], Definition 2.1, (iii)] for the subset of Edge(G 0 | v ) (= Cusp(G 0 | v )) consisting of cusps of G 0 | v that arise from nodes of G 0 . (iii) We shall write Aut |E(G| v :G)| (G 0 | v ) Glu brch (G 0 ) ⊆ v∈Vert(G 0 ) [cf. (ii); [CbTpI], Definition 2.6, (i)] for the [closed] subgroup of |E(G| v :G)| (G 0 | v ) consisting of “glueable” collections v∈Vert(G 0 ) Aut 104 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI of automorphisms of the various G 0 | v , i.e., the subgroup con- sisting of (α v ) v∈Vert(G 0 ) such that, for every v, w ∈ Vert(G 0 ), it holds that χ v (α v ) = χ w (α w ) [cf. [CbTpI], Definition 3.8, (ii)]. Remark 4.1.1. In the notation of Definition 4.1, one verifies easily from the various definitions involved that   brch |grph| Glu(G 0 ) = Glu (G 0 ) ∩ Aut (G 0 | v ) v∈Vert(G 0 ) [cf. [CbTpI], Definitions 2.6, (i), and 4.9; Remark 4.1.2 of the present monograph]. Remark 4.1.2. Here, we take the opportunity to correct a minor error in the exposition of [CbTpI]. In [CbTpI], Definition 2.6, (i), “Aut |grph| (G)” should be defined as the subgroup of Aut(G) of automor- phisms of G which induce the identity automorphism on the underlying semi-graph of G [cf. the definition given in [CbTpI], Theorem B]. In a similar vein, in [CbTpI], Definition 2.6, (iii), “Aut |H| (G)” should be de- fined as the subgroup of Aut(G) of automorphisms of G which preserve the sub-semi-graph H of the underlying semi-graph of G and, moreover, induce the identity automorphism of H. Since the correct definitions are applied throughout the exposition of [CbTpI], these errors in the statement of the definitions have no substantive effect on the exposition of [CbTpI], except for the following two instances [which themselves do not have any substantive effect on the exposition of [CbTpI]]: (i) In [CbTpI], Proposition 2.7, (ii), “Aut |grph| (G)” should be re- placed by “Aut |VCN(G)| (G)”. (ii) In [CbTpI], Proposition 2.7, (iii), the phrase “In particular” should be replaced by the word “Finally”. Theorem 4.2 (Glueability of combinatorial cuspidalizations in the one-dimensional case). Let Σ 0 be a nonempty set of prime num- bers and G 0 a semi-graph of anabelioids of pro-Σ 0 PSC-type. Write Π G 0 for the [pro-Σ 0 ] fundamental group of G 0 . Then the following hold: (i) The closed subgroup Dehn(G 0 ) ⊆ Aut(G 0 ) [cf. [CbTpI], Def- inition 4.4] is contained in Aut |Brch(G 0 )| (G 0 ) ⊆ Aut(G 0 ) [cf. Definition 4.1, (i)], i.e., Dehn(G 0 ) ⊆ Aut |Brch(G 0 )| (G 0 ). (ii) The natural homomorphism Aut |Brch(G 0 )| (G 0 ) −→ α → v∈Vert(G 0 ) Aut(G 0 | v ) (α G 0 | v ) v∈Vert(G 0 ) COMBINATORIAL ANABELIAN TOPICS II 105 [cf. [CbTpI], Definition 2.14, (ii); [CbTpI], Remark 2.5.1, (ii)] factors through Glu brch (G 0 ) ⊆ Aut(G 0 | v ) v∈Vert(G 0 ) [cf. Definition 4.1, (iii)]. (iii) The natural inclusion Dehn(G 0 ) → Aut |Brch(G 0 )| (G 0 ) of (i) and |Brch(G 0 )| the natural homomorphism ρ brch (G 0 ) → Glu brch (G 0 ) G 0 : Aut [cf. (ii)] fit into an exact sequence of profinite groups ρ brch G 0 1 −→ Dehn(G 0 ) −→ Aut |Brch(G 0 )| (G 0 ) −→ Glu brch (G 0 ) −→ 1 . Proof. Assertion (i) follows immediately from the various definitions involved. Assertion (ii) follows immediately from [CbTpI], Corollary 3.9, (iv). Assertion (iii) follows, in light of Remark 4.1.1, from the exact sequence of [CbTpI], Theorem B, (iii), together with the existence of automorphisms of G 0 that induce arbitrary permutations of the cusps on each vertex of G 0 and, moreover, restrict to automorphisms of each G 0 | v that lie in the kernel of χ v [cf. the automorphisms constructed in the proof of [CmbCsp], Lemma 2.4].  Definition 4.3. Let H be a sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of G [cf. Definition 3.1, (ii)] and S ⊆ Node(G| H ) [cf. [CbTpI], Definition 2.2, (ii)] a subset of Node(G| H ) that is not of separating type [cf. [CbTpI], Definition 2.5, (i)]. Then, by applying a similar argument to the argument applied in [CmbCsp], Definition 2.1, (iii), (vi), or [NodNon], Definition 5.1, (ix), (x) [i.e., by considering the portion of the underlying scheme X n of X n log corresponding to the underlying scheme (X H,S ) n of the n-th log configuration space (X H,S ) log n log of the stable log curve X H,S determined by (G| H ) S — cf. [CbTpI], Definition 2.5, (ii)], one obtains a closed subgroup (Π H,S ) n ⊆ Π n [which is well-defined up to Π n -conjugation]. We shall refer to (Π H,S ) n ⊆ Π n as a configuration space subgroup [associated to (H, S)]. For each 0 ≤ i ≤ j ≤ n, we shall write def (Π H,S ) n/i = (Π H,S ) n ∩ Π n/i ⊆ Π n/i [which is well-defined up to Π n -conjugation]; def (Π H,S ) j/i = (Π H,S ) n/i /(Π H,S ) n/j ⊆ Π j/i [which is well-defined up to Π j -conjugation]. In particular, (Π H,S ) j = (Π H,S ) j/0 ⊆ Π j 106 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [where we recall that, in fact, the subgroups on either side of the “=” are only well-defined up to Π j -conjugation]. Thus, by applying [CbTpI], Proposition 2.11, inductively, we conclude that each (Π H,S ) j/i is a pro-Σ configuration space group [cf. [MzTa], Definition 2.3, (i)], and that we have a natural exact sequence of profinite groups 1 −→ (Π H,S ) j/i −→ (Π H,S ) j −→ (Π H,S ) i −→ 1 . Finally, let v ∈ Vert(G). Then the semi-graph of anabelioids of PSC- type G| v [cf. [CbTpI], Definition 2.1, (iii)] may be naturally identified with (G| H v ) S v for suitable choices of H v , S v [cf. [CbTpI], Remark 2.5.1, (ii)]. We shall refer to def (Π v ) n = (Π H v ,S v ) n ⊆ Π n as a configuration space subgroup associated to v. Thus, (Π v ) 1 ⊆ Π 1 is a verticial subgroup associated to v ∈ Vert(G), i.e., a subgroup that is typically denoted “Π v ”. We shall write def (Π v ) j/i = (Π H v ,S v ) j/i ⊆ Π j/i . Remark 4.3.1. In the notation of Definition 4.3, one verifies easily — by applying a suitable specialization isomorphism [cf. the discus- sion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] — that there exist a stable log curve Y log over (Spec k) log and an n-cuspidalizable degeneration structure (G, S, φ) on Y G [cf. Defini- tion 3.23, (i), (v)] — where we write Y G for the “G” that occurs in the case where we take “X log ” to be Y log — which satisfy the following: Write Y Π n for the “Π n ” that occurs in the case where we take “X log ” to be Y log . Then: The image of a configuration space subgroup of Π n associated to (H, S) [cf. Definition 4.3] via a PFC- ∼ admissible outer isomorphism Π n → Y Π n that lies over the displayed composite isomorphism of Definition 3.23, (v) [where we note that, in loc. cit., the roles of “ Y Π n ” and “Π n ” are reversed!], is a configuration space sub- group of Y Π n associated to a vertex of Y G. Lemma 4.4 (Commensurable terminality and slimness). Every configuration space subgroup [cf. Definition 4.3] of Π n is topo- logically finitely generated, slim, and commensurably terminal in Π n . Proof. Since any configuration space subgroup is, in particular, a con- figuration space group, the fact that such a subgroup is topologically finitely generated and slim follows from [MzTa], Proposition 2.2, (ii). COMBINATORIAL ANABELIAN TOPICS II 107 Thus, it remains to verify commensurable terminality. By applying the observation of Remark 4.3.1, we reduce immediately to the case of a configuration space subgroup associated to a vertex. But then the desired commensurable terminality follows, in light of Lemma 4.5 below, by induction on n, together with the corresponding fact for n = 1 [cf. [CmbGC], Proposition 1.2, (ii)]. This completes the proof of Lemma 4.4.  Lemma 4.5 (Extensions and commensurable terminality). Let 1 −−−→ N H −−−→ H −−−→ Q H −−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    1 −−−→ N −−−→ G −−−→ Q −−−→ 1 be a commutative diagram of profinite groups, where the horizontal se- quences are exact, and the vertical arrows are injective. Suppose that N H ⊆ N , Q H ⊆ Q are commensurably terminal in N , Q, respectively. Then H ⊆ G is commensurably terminal in G.  Proof. This follows immediately from Lemma 3.9, (i). Definition 4.6. (i) We shall write Out FC (Π n ) brch ⊆ Out FC (Π n ) for the closed subgroup of Out FC (Π n ) given by the inverse im- age of ∼ Aut |Brch(G)| (G) ⊆ (Aut(G) ⊆) Out(Π G ) ← Out(Π 1 ) [cf. Definition 4.1, (i)] via the natural injection Out FC (Π n ) → Out FC (Π 1 ) ⊆ Out(Π 1 ) of [NodNon], Theorem B. def (ii) Let v ∈ Vert(G); write Π v = (Π v ) 1 [cf. Definition 4.3]. Then we shall write Out FC ((Π v ) n ) G-node ⊆ Out FC ((Π v ) n ) for the [closed] subgroup of Out FC ((Π v ) n ) given by the inverse image of Aut |E(G| v :G)| (G| v ) ⊆ (Aut(G| v ) ⊆) Out(Π v ) [cf. Definition 4.1, (ii); [CbTpI], Definition 2.6, (i)] via the natural injection Out FC ((Π v ) n ) → Out FC (Π v ) ⊆ Out(Π v ) of [NodNon], Theorem B. 108 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Theorem 4.7 (Graphicity of outomorphisms of certain subquo- tients). In the notation of the preceding §3 [cf. also Definition 3.1], let x ∈ X n (k). Write C x ⊆ Cusp(G) for the [possibly empty] set consisting of cusps c of G such that, for some i ∈ {1, · · · , n}, x {i} ∈ X {i} (k) = X(k) [cf. Definition 3.1, (i)] lies on the cusp of X log corresponding to c ∈ Cusp(G). For each i ∈ {1, · · · , n}, write def G i/i−1,x = G i∈{1,··· ,i},x [cf. Definition 3.1, (iii)] and z i/i−1,x ∈ VCN(G i/i−1,x ) for the element of VCN(G i/i−1,x ) on which x {1,··· ,i} lies, that is to say: If x {1,··· ,i} ∈ X i (k) [cf. the notation given in the discus- sion preceding Definition 3.1] is a cusp or node of the log log → X i−1 geometric fiber of the projection p log i/i−1 : X i over x log {1,··· ,i−1} corresponding to an edge e ∈ Edge(G i/i−1,x ), def then z i/i−1,x = e; if x {1,··· ,i} ∈ X i (k) is neither a cusp nor node of the geometric fiber of the projection log log p log → X i−1 over x log i/i−1 : X i {1,··· ,i−1} but lies on the irre- ducible component of the geometric fiber corresponding def to a vertex v ∈ Edge(G i/i−1,x ), then z i/i−1,x = v. Let α ∈ Out FC (Π n ) brch [cf. Definition 4.6, (i)]. Suppose that the element of ∼ Aut |Brch(G)| (G) ⊆ (Aut(G) ⊆) Out(Π G ) ← Out(Π 1 ) [cf. Definition 4.1, (i)] determined by α ∈ Out FC (Π n ) brch [cf. Defini- tion 4.6, (i)] is contained in Aut |C x | (G) ⊆ Aut(G) [cf. [CbTpI], Definition 2.6, (i)]. Then there exist • a lifting α  ∈ Aut(Π n ) of α, and, ∼ • for each i ∈ {1, · · · , n}, a VCN-subgroup Π z i/i−1,x ⊆ Π i/i−1 → Π G i/i−1 ,x [cf. Definition 3.1, (iii)] associated to the element z i/i−1,x ∈ VCN(G i/i−1,x ) such that the following properties hold: ∼ (a) For each i ∈ {1, · · · , n}, the automorphism of Π i/i−1 → Π G i/i−1,x ∼ determined by α  fixes the VCN-subgroup Π z i/i−1,x ⊆ Π i/i−1 → Π G i/i−1,x . COMBINATORIAL ANABELIAN TOPICS II 109 ∼ (b) For each i ∈ {1, · · · , n}, the outomorphism of Π i/i−1 → Π G i/i−1,x induced by α  is contained in ∼ Aut |Brch(G i/i−1,x )| (G i/i−1,x ) ⊆ Out(Π G i/i−1,x ) ← Out(Π i/i−1 ) . Proof. We verify Theorem 4.7 by induction on n. If n = 1, then Theo- rem 4.7 follows immediately from the various definitions involved. Now suppose that n ≥ 2, and that the induction hypothesis is in force. In particular, [since the homomorphism p Π n/n−1 : Π n  Π n−1 is surjective] we have a lifting α  ∈ Aut(Π n ) of α and, for each i ∈ {1, · · · , n − 1}, ∼ a VCN-subgroup Π z i/i−1,x ⊆ Π i/i−1 → Π G i/i−1 ,x associated to the ele- ment z i/i−1,x ∈ VCN(G i/i−1,x ) such that, for each i ∈ {1, · · · , n − 1}, the automorphism of Π i determined by α  fixes Π z i/i−1,x ⊆ Π i/i−1 ⊆ Π i , and, moreover, the automorphism of Π n−1 determined by α  satisfies the property (b) in the statement of Theorem 4.7. Now we claim that the following assertion holds: ∼ Claim 4.7.A: The outomorphism of Π n/n−1 → Π G n/n−1,x induced by the lifting α  is contained in ∼ Aut |Brch(G n/n−1,x )| (G n/n−1,x ) ⊆ Out(Π G n/n−1,x ) ← Out(Π n/n−1 ) . To this end, let us first observe that it follows immediately — by re- log log → X n−2 via a suit- placing X n log by the base-change of p log n/n−2 : X n log able morphism of log schemes (Spec k) log → X n−2 whose image lies on x {1,··· ,n−2} ∈ X n−2 (k) — from Lemma 3.2, (iv), that, to verify Claim 4.7.A, we may assume without loss of generality that n = 2. Also, one verifies easily, by applying Lemma 3.14, (i) [cf. also [CbTpI], Proposi- tion 2.9, (i)], and possibly replacing, when z 1/0,x ∈ Vert(G 1/0,x ), • α  by the composite of α  with an inner automorphism of Π n = Π 2 determined by conjugation by a suitable element of Π n = ∼ Π 2 whose image in Π 1 → Π G 1/0,x is contained in the closed ∼ subgroup Π z 1/0,x ⊆ Π G 1/0,x ← Π 1 and • x by a suitable “x” whose associated “z 1/0,x ” is a node of G 1/0,x that abuts to the original z 1/0,x ∈ Vert(G 1/0,x ), that we may assume without loss of generality that z 1/0,x ∈ Edge(G 1/0,x ). ∼ Next, let us recall that the automorphism of Π 1 → Π G 1/0,x determined ∼ by α  fixes the edge-like subgroup Π z 1/0,x ⊆ Π 1 → Π G 1/0,x associated to the edge z 1/0,x of G 1/0,x [cf. the discussion preceding Claim 4.7.A]. Thus, since [we have assumed that] α ∈ Out FC (Π 2 ) brch [which implies ∼ that the outomorphism of Π 1 → Π G 1/0,x determined by α preserves the ∼ Π 1 -conjugacy class of each verticial subgroup of Π 1 → Π G 1/0,x ], it fol- lows immediately from Lemma 3.13, (i), (ii), that the outomorphism ∼ of Π G 2/1,x ← Π 2/1 induced by α  is group-theoretically verticial, hence [cf. [NodNon], Proposition 1.13; [CmbGC], Proposition 1.5, (ii); the fact that α is C-admissible] graphic, i.e., ∈ Aut(G 2/1,x ). Moreover, since 110 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ the outomorphism of Π G 2∈{2},x ← Π 1 induced by α  is, by assumption, |Brch(G)| (G) [cf. [CmbCsp], Proposition 1.2, (iii)], one contained in Aut verifies easily, by considering the map on vertices/nodes/branches in- duced by the projection p Π {1,2}/{2} | Π 2/1 : Π 2/1  Π {2} ∼ [cf. Lemma 3.6, (i), (iv)], that the outomorphism of Π G 2/1,x ← Π 2/1 induced by α  is contained in the subgroup Aut |Brch(G 2/1,x )| (G 2/1,x ). This completes the proof of Claim 4.7.A. On the other hand, one verifies easily from Claim 4.7.A, together with the various definitions involved, that there exist a Π n/n−1 -conjugate β  ∼ of α  and a VCN-subgroup Π z n/n−1 ,x ⊆ Π n/n−1 → Π G n/n−1,x associated to z n/n−1,x ∈ VCN(G n/n−1,x ) such that β  fixes Π z n/n−1 ,x . In particular, the lifting β  of α and the VCN-subgroups Π z i/i−1 ,x [where i ∈ {1, · · · , n}] satisfy the properties (a), (b) in the statement of Theorem 4.7. This completes the proof of Theorem 4.7.  Lemma 4.8 (Preservation of configuration space subgroups). The following hold: (i) Let α ∈ Out FC (Π n ) brch [cf. Definition 4.6, (i)]. Then α pre- serves the Π n -conjugacy class of each configuration space sub- group [cf. Definition 4.3] of Π n . Thus, by applying the portion of Lemma 4.4 concerning commensurable terminality, to- gether with Lemma 3.10, (i), we obtain a natural homomor- phism Out FC (Π n ) brch −→ Out((Π v ) n ) . v∈Vert(G) (ii) The displayed homomorphism of (i) factors through Out FC ((Π v ) n ) G-node ⊆ v∈Vert(G) Out((Π v ) n ) v∈Vert(G) [cf. Definition 4.6, (ii)]. Proof. First, we verify assertion (i). We begin by observing that, in light of the observation of Remark 4.3.1 [cf. also [CbTpI], Proposition 2.9, (ii)], to complete the verification of assertion (i), it suffices to verify the following assertion: Claim 4.8.A: For each v ∈ Vert(G), α preserves the Π n -conjugacy class of configuration space subgroups (Π v ) n ⊆ Π n of Π n associated to v. COMBINATORIAL ANABELIAN TOPICS II 111 To verify Claim 4.8.A, let us observe that, by applying Theorem 4.7 in the case where we take the “x” in the statement of Theorem 4.7 to be such that, for each i ∈ {1, · · · , n}, the element z i/i−1,x ∈ Vert(G i/i−1,x ) is the vertex of G i/i−1,x that corresponds [via the various bijections of Lemma 3.6, (iii)] to the vertex v of Claim 4.8.A, we obtain, for each ∼ i ∈ {1, · · · , n}, a VCN-subgroup Π z i/i−1,x ⊆ Π i/i−1 → Π G i/i−1 ,x associ- ated to z i/i−1,x ∈ VCN(G i/i−1,x ) as in the statement of Theorem 4.7, (a). Next, let us observe that one verifies immediately from the com- mensurable terminality [cf. [CmbGC], Proposition 1.2, (ii)] of each of the VCN-subgroups Π z i/i−1,x ⊆ Π i/i−1 , where i ∈ {1, · · · , n}, that the Π n -conjugacy class of the configuration space subgroup (Π v ) n ⊆ Π n coincides with the Π n -conjugacy class of the closed subgroup of Π n consisting of γ ∈ Π n such that, for each i ∈ {1, · · · , n}, conjuga- tion by γ preserves the closed subgroup Π z i/i−1,x ⊆ (Π i/i−1 ⊆) Π i [so Π z i/i−1,x = (Π v ) i/i−1 ]. Thus, it follows from Theorem 4.7, (a), that α preserves the Π n -conjugacy class of (Π v ) n ⊆ Π n , as desired. This completes the proof of Claim 4.8.A. Next, we verify assertion (ii). Let α ∈ Out FC (Π n ) brch , v ∈ Vert(G). Write α v for the outomorphism of (Π v ) n induced by α [cf. (i)]. Then the F-admissibility of α v follows immediately from the F-admissibility of α [cf. the discussion of Definition 4.3]. The C-admissibility of α v follows immediately from Theorem 4.7 [applied as in the proof of Claim 4.8.A]; [NodNon], Lemma 1.7, together with the definition of C-admissibility. Finally, the fact that α v ∈ Out FC ((Π v ) n ) G-node follows immediately from the fact that α ∈ Out FC (Π n ) brch . This completes the proof of assertion (ii).  Definition 4.9. We shall write Out FC ((Π v ) n ) G-node Glu(Π n ) ⊆ v∈Vert(G) for the [closed] subgroup of v∈Vert(G) Out FC ((Π v ) n ) G-node consisting of “glueable” collections of outomorphisms of the various (Π v ) n , i.e., the subgroup defined as follows: (i) Suppose that n = 1. Then Glu(Π n ) consists of those collections (α v ) v∈Vert(G) such that, for every v, w ∈ Vert(G), it holds that χ v (α v ) = χ w (α w ) [cf. [CbTpI], Definition 3.8, (ii)] — where we note that one verifies easily that α v may be regarded as an element of Aut(G| v ). (ii) Suppose that n = 2. Then Glu(Π n ) consists of those collections (α v ) v∈Vert(G) that satisfy the following condition: Let v, w ∈ Vert(G); e ∈ N (v) ∩ N (w); T ⊆ Π 2/1 ⊆ Π 2 = Π n a {1, 2}- tripod of Π n arising from e ∈ N (v) ∩ N (w) [cf. Definitions 112 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 3.3, (i); 3.7, (i)]. Then one verifies easily from the various definitions involved that there exist Π n -conjugates T v , T w of T such that T v , T w are contained in (Π v ) n , (Π w ) n , respectively, and, moreover, T v ⊆ (Π v ) 2/1 ⊆ (Π v ) 2 = (Π v ) n , T w ⊆ (Π w ) 2/1 ⊆ (Π w ) 2 = (Π w ) n are tripods of (Π v ) n , (Π w ) n arising from [the cusps of G| v , G| w corresponding to] the node e, respectively. Moreover, since α v ∈ Out FC ((Π v ) n ) G-node , α w ∈ Out FC ((Π w ) n ) G-node , it follows from Theorem 3.16, (iv), that α v ∈ Out FC ((Π v ) n )[T v ], α w ∈ ∼ Out FC ((Π w ) n )[T w ]; thus, we obtain that T T v (α v ) ∈ Out(T v ) → ∼ Out(T ); T T w (α w ) ∈ Out(T w ) → Out(T ) [cf. Theorem 3.16, (i)]. Then we require that T T v (α v ) = T T w (α w ). (iii) Suppose that n ≥ 3. Then Glu(Π n ) consists of those col- lections (α v ) v∈Vert(G) that satisfy the following condition: Let Π tpd ⊆ Π 3 be a 3-central {1, 2, 3}-tripod of Π n [cf. Definitions 3.3, (i); 3.7, (ii)]. Then one verifies easily from the various definitions involved that, for every v ∈ Vert(G), there exists a of Π tpd such that Π tpd is contained in (Π v ) 3 , Π 3 -conjugate Π tpd v v tpd and, moreover, Π v ⊆ (Π v ) 3 is a 3-central tripod of (Π v ) 3 . (α v ) ∈ Thus, since α v ∈ Out FC ((Π v ) n ) G-node , we obtain T Π tpd v tpd ∼ tpd Out(Π v ) → Out(Π ) [cf. Theorem 3.16, (i), (v)]. Then, for (α v ) = T Π tpd (α w ). every v, w ∈ Vert(G), we require that T Π tpd v w Remark 4.9.1. In the notation of Definition 4.9, one verifies eas- ily from the various definitions involved that the natural outer iso- ∼ ∼ moprhism Π 1 → Π G determines a natural isomorphism Glu(Π 1 ) → Glu brch (G) [cf. Definition 4.1, (iii)]. Lemma 4.10 (Basic properties concerning groups of glueable collections). For n ≥ 1, the following hold: (i) The natural injections Out FC ((Π v ) n+1 ) → Out FC ((Π v ) n ) of [NodNon], Theorem B — where v ranges over the vertices of G — determine an injection Glu(Π n+1 ) → Glu(Π n ) . COMBINATORIAL ANABELIAN TOPICS II 113 (ii) The displayed homomorphism of Lemma 4.8, (i), Out FC (Π n ) brch −→ Out((Π v ) n ) v∈Vert(G) factors through Glu(Π n ) ⊆ Out((Π v ) n ) . v∈Vert(G) Proof. First, we verify assertion (i). The fact that the image of the composite Glu(Π n+1 ) → Out FC ((Π v ) n+1 ) → v∈Vert(G) Out FC ((Π v ) n ) v∈Vert(G) is contained in Out FC ((Π v ) n ) G-node ⊆ v∈Vert(G) Out FC ((Π v ) n ) v∈Vert(G) follows immediately from the various definitions involved. The fact that the image of the composite Glu(Π n+1 ) → Out FC ((Π v ) n+1 ) → v∈Vert(G) Out FC ((Π v ) n ) v∈Vert(G) is contained in Out FC ((Π v ) n ) G-node Glu(Π n ) ⊆ v∈Vert(G) follows immediately from the various definitions involved when n ≥ 3 and from Theorems 3.16, (iv), (v); 3.18, (ii) [applied to each (Π v ) n+1 !], when n = 2. Thus, it remains to verify assertion (i) in the case where n = 1. Suppose that n = 1. Let (α v ) v∈Vert(G) ∈ Glu(Π 2 ). Write ((α v ) 1 ) v∈Vert(G) ∈ v∈Vert(G) Out FC ((Π v ) 1 ) G-node for the image of (α v ) v∈Vert(G) . Since G is connected, to verify assertion (i) in the case where n = 1, it suffices to verify that, for any two vertices v, w of G such that N (v) ∩ N (w)  = ∅, it holds that χ v ((α v ) 1 ) = χ w ((α w ) 1 ). Let x ∈ X 2 (k) be a k-valued geometric point of X 2 such that x {1} ∈ X(k) [cf. Definition 3.1, (i)] is a node of X log corresponding to an element of N (v) ∩ N (w)  = ∅. Then by applying Theorem 4.7 to a suitable lifting α  v (∈ Aut FC ((Π v ) 2 )) of the outomorphism α v of (Π v ) 2 [where we take the “Π n ” in the statement of Theorem 4.7 to be (Π v ) 2 ], we ∼ conclude that the outomorphism (α v ) 2/1 of Π (G| v ) 2∈{1,2},x ← (Π v ) 2/1 [cf. Definition 3.1, (iii)] determined by α  v is graphic and fixes each of the vertices of (G| v ) 2∈{1,2},x . Thus, if we write (α v ) {2} for the outomorphism of the “Π {2} ” that occurs in the case where we take “Π 2 ” to be (Π v ) 2 , then it follows from [CmbCsp], Proposition 1.2, (iii), together with the C-admissibility of (α v ) 1 , that (α v ) {2} is C-admissible, i.e., ∈ Aut(G| v ). 114 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Now, for a [{1, 2}-]tripod T v ⊆ (Π v ) 2 arising from the cusp x {1} of G| v [cf. Definitions 3.3, (i); 3.7, (i)], we compute: χ G| v ((α v ) {2} ) [cf. [CmbCsp], Proposition 1.2, (iii)] χ G| v ((α v ) 1 ) = = χ (G| v ) 2∈{1,2},x ((α v ) 2/1 ) [cf. [CbTpI], Corollary 3.9, (iv)] [cf. [CbTpI], Corollary 3.9, (iv)] = χ T v ((α v ) 2/1 | T v ) [where we refer to Lemma 3.12, (i), concerning “(α v ) 2/1 | T v ”, and we write χ T v for the “χ” associated to the vertex of (G| v ) 2∈{1,2},x corre- sponding to T v ]. Moreover, by applying a similar argument to the above argument, we conclude that there exists a lifting α  w of α w such that ∼  w the outomorphism (α w ) 2/1 of Π (G| w ) 2∈{1,2},x ← (Π w ) 2/1 determined by α is graphic [and fixes each of the vertices of (G| w ) 2∈{1,2},x ], and, more- over, for a [{1, 2}-]tripod T w ⊆ (Π w ) 2 arising from the cusp x {1} of G| w , it holds that χ G| w ((α w ) 1 ) = χ T w ((α w ) 2/1 | T w ). On the other hand, since (α v ) v∈Vert(G) ∈ Glu(Π 2 ), it holds that χ T v ((α v ) 2/1 | T v ) = χ T w ((α w ) 2/1 | T w ). In particular, we obtain that χ G| v ((α v ) 1 ) = χ G| w ((α w ) 1 ). This completes the proof of assertion (i). Next, we verify assertion (ii). If n = 1, then assertion (ii) amounts to Theorem 4.2, (ii) [cf. also Remark 4.9.1]. If n ≥ 2, then assertion (ii) follows immediately from Lemma 4.8, (ii), together with the fact that the homomorphism “T T ” of Theorem 3.16, (i), does not depend on the choice of “T ” among its conjugates. This completes the proof of assertion (ii).  Definition 4.11. We shall write ρ brch for the homomorphism n Out FC (Π n ) brch −→ Glu(Π n ) determined by the factorization of Lemma 4.10, (ii). Lemma 4.12 (Glueable collections in the case of precisely one node). Suppose that n = 2, and that #Node(G) = 1. Let v  , w  ∈  Vert( G) be distinct elements such that N ( v ) ∩ N ( w)   = ∅. Write e  ∈  Node( G) for the unique element of N ( v )∩N ( w)  [cf. [NodNon], Lemma ∼ ∼ 1.8]; Π v  , Π w  , Π e  ⊆ Π G ← Π 1 for the VCN-subgroups of Π G ← Π 1 as- def  respectively; v def sociated to v  , w,  e  ∈ VCN( G), = v  (G); w = w(G);  def e = e  (G). [Thus, one verifies easily that Π e  = Π v  ∩ Π w  [cf. [NodNon], Lemma 1.9, (i)], that Vert(G) = {v, w}, and that if G is noncycli- cally primitive (respectively, cyclically primitive) [cf. [CbTpI], Definition 4.1], then v  = w (respectively, v = w).] Let x ∈ X 2 (k) be a k-valued geometric point of X 2 such that x {1} ∈ X(k) [cf. Defini- tion 3.1, (i)] lies on the unique node of X log [i.e., which corresponds def to e]. Write G 2/1 = G 2∈{1,2},x [cf. Definition 3.1, (iii)]; G  2/1 → G 2/1 COMBINATORIAL ANABELIAN TOPICS II 115 ∼ for the profinite étale covering corresponding to Π G 2/1 ← Π 2/1 ; v new new for the “v 2,1,x ” of Lemma 3.6, (iv). For each z ∈ Vert(G), write ◦ z ∈ Vert(G 2/1 ) for the vertex of G 2/1 that corresponds to z via the bijections of Lemma 3.6, (i), (iv). [Thus, it follows from Lemma 3.6, (iv), that Vert(G 2/1 ) = {v new , v ◦ , w ◦ }.] Then the following hold [cf. also Figures 2, 3, below]: (i) Let (Π v  ) 2 ⊆ Π 2 be a configuration space subgroup of Π 2 as- sociated to v [cf. Definition 4.3] such that the image of the p Π 2/1 ∼ composite (Π v  ) 2 → Π 2  Π 1 coincides with Π v  ⊆ Π G ← Π 1 . ∼ Also, let us fix a verticial subgroup Π v  new ⊆ Π G 2/1 ← Π 2/1 of ∼ Π G 2/1 ← Π 2/1 associated to a v  new ∈ Vert( G  2/1 ) that lies over v new ∈ Vert(G 2/1 ) and is contained in (Π v  ) 2 . Then there exists a unique configuration space subgroup (Π w  ) 2 ⊆ Π 2 of Π 2 associated to w [cf. Definition 4.3] such that Π v  new = def (Π v  ) 2/1 ∩ (Π w  ) 2/1 — where we write (Π v  ) 2/1 = Π 2/1 ∩ (Π v  ) 2 ; def (Π w  ) 2/1 = Π 2/1 ∩ (Π w  ) 2 — and, moreover, the image of the p Π 2/1 composite (Π w  ) 2 → Π 2  Π 1 coincides with Π w  ⊆ Π 1 . (ii) In the situation of (i), the natural homomorphism lim (Π v  ← Π e  → Π w  ) −→ Π 1 −→ — where the inductive limit is taken in the category of pro- Σ groups — is injective, and its image is commensurably terminal in Π 1 . Write Π v  , w  ⊆ Π 1 for the image of the above p Π 2/1 homomorphism; Π 2 | Π v  , w  (⊆ Π 2 ) for the fiber product of Π 2  Π 1 and Π v  , w  → Π 1 . Thus, we have an exact sequence of profinite groups 1 −→ Π 2/1 −→ Π 2 | Π v  , w  −→ Π v  , w  −→ 1 . Finally, if G is noncyclically primitive, then Π v  , w  = Π 1 , Π 2 | Π v  , w  = Π 2 . ∼ (iii) In the situation of (ii), for each z  ∈ { v , w},  let Π z  ◦ ⊆ Π G 2/1 ← ∼ Π 2/1 be a verticial subgroup of Π G 2/1 ← Π 2/1 associated to a z  ◦ ∈ Vert( G  2/1 ) that lies over z ◦ ∈ Vert(G 2/1 ) such that Π z  ◦ ⊆ (Π z  ) 2/1 [cf. (i)], and, moreover, Π z  ◦ ∩ Π v  new  = {1}. Thus, ∼ def Π e  z  ◦ = Π z  ◦ ∩ Π v  new is the nodal subgroup of Π G 2/1 ← Π 2/1 associated to the unique element e  z  ◦ of N ( z ◦ ) ∩ N ( v new ) [cf. def [NodNon], Lemma 1.9, (i)]. Write e z ◦ = e  z  ◦ (G 2/1 ). Then the natural homomorphism lim (Π z  ◦ ← Π e  z  ◦ → Π v  new ) −→ (Π z  ) 2/1 −→ 116 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI G 2/1 Π H v  ◦ Π H w  ◦ || || (Π v  ) 2/1 (Π w  ) 2/1      e  v  ◦ e  w  ◦ ··· • • • ··· ◦ new · · v  · v  w  ◦ ·  · · · · · · · · ∨ • v ◦ ↓ · · · · · · · · ∨ • • new e v ◦ v e w ◦ w ◦ G = G 1  ··· • v  Π v  , w    • ··· e  w  ↓ • v e • w Figure 2 : the noncyclically primitive case COMBINATORIAL ANABELIAN TOPICS II G 2/1 Π H v  ◦ 117 Π H w  ◦ || || (Π v  ) 2/1 (Π w  ) 2/1      e  v  ◦ e  w  ◦ • • • ◦ · new · v  · v  w  ◦ ·  ··· · · · · · · · · · · · · · · e v ◦ · > G = G 1 ···  • v  · · · · ↓ · · · · new · · v · · • · · < · • ◦ v = w ◦ e w ◦ Π v  , w    • e  w  ··· ↓ e • v = w Figure 3 : the cyclically primitive case ··· 118 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI — where the inductive limit is taken in the category of pro- Σ groups — is an isomorphism. Write G † z ◦ for the sub- semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of the underlying semi-graph of G 2/1 whose set of vertices = def { z (G) ◦ , v new }; T z ◦ = (Node(G 2/1 ) \ {e z ◦ }) ∩ Node(G 2/1 | G † ◦ ) ⊆ z Node(G 2/1 ) [cf. [CbTpI], Definition 2.2, (ii)]. Then the natu- ral homomorphism of the above display allows one to identify (Π z  ) 2/1 with the [pro-Σ] fundamental group Π H z  ◦ of def H z ◦ = (G 2/1 | G † ◦ ) T z ◦ z [cf. [CbTpI], Definition 2.5, (ii)]. (iv) In the situation of (iii), let (α z ) z∈Vert(G) ∈ Glu(Π 2 ). Write ((α z ) 1 ) z∈Vert(G) ∈ Glu(Π 1 ) for the image of (α z ) z∈Vert(G) ∈ Glu(Π 2 ) via the injection of Lemma 4.10, (i). Let α 1 ∈ Aut |Brch(G)| (G) be such that ρ brch (α 1 ) = ((α z ) 1 ) z∈Vert(G) ∈ Glu(Π 1 ) [cf. Theo- 1 rem 4.2, (iii); Definition 4.11]. Then the outomorphism α 1 of Π 1 preserves the Π 1 -conjugacy class of Π v  , w  ⊆ Π 1 . Thus, by applying the portion of (ii) concerning commensurable termi- nality, we obtain [cf. Lemma 3.10, (i)] a restricted outomor- phism α 1 | Π v  , w  ∈ Out(Π v  , w  ). (v) In the situation of (iv), there exists an outomorphism β v  , w  [α 1 ] of Π 2 | Π v  , w  that satisfies the following conditions: (1) β v  , w  [α 1 ] preserves Π 2/1 ⊆ Π 2 | Π v  , w  and the Π 2 | Π v  , w  -conjugacy classes of (Π v  ) 2 , (Π w  ) 2 ⊆ Π 2 | Π v  , w  . (2) There exists an automorphism β  v  , w  [α 1 ] of Π 2 | Π v  , w  that lifts the outomorphism β v  , w  [α 1 ] such that the outomorphism of ∼ Π G 2/1 ← Π 2/1 determined by β  v  , w  [α 1 ] [cf. (1)] is con- ∼ tained in Aut |Brch(G 2/1 )| (G 2/1 ) ⊆ Out(Π G 2/1 ) ← Out(Π 2/1 ). (3) For each z  ∈ { v , w},  the outomorphism β v  , w  [α 1 ]| (Π z  ) 2 of (Π z  ) 2 determined by β v  , w  [α 1 ] [i.e., obtained by applying (1) and Lemma 3.10, (i) — where we note that (Π z  ) 2 is commensurably terminal in Π 2 [cf. Lemma 4.4], hence also in Π 2 | Π v  , w  ] coincides with α z  (G) [cf. the notation of (iv)]. (4) The outomorphism of Π v  , w  induced by β v  , w  [α 1 ] [cf. (1)] coincides with α 1 | Π v  , w  [cf. (iv)]. Here, we observe, in the context of (2), that the outer isomor- ∼ phism Π 2/1 → Π G 2/1 [i.e., which gives rise to “the” closed sub- ∼ group Aut |Brch(G 2/1 )| (G 2/1 ) ⊆ Out(Π G 2/1 ) ← Out(Π 2/1 )] may be characterized, up to composition with elements of the subgroup COMBINATORIAL ANABELIAN TOPICS II 119 ∼ Aut |Brch(G 2/1 )| (G 2/1 ) ⊆ Out(Π G 2/1 ) ← Out(Π 2/1 ), as the group- theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)] outer isomorphism such that the semi-graph of anabelioids structure on G 2/1 is the semi-graph of anabelioids structure determined [cf. [NodNon], Theorem A] by the resulting composite ∼ ∼ Π e  → Π G ← Π 1 → Out(Π 2/1 ) → Out(Π G 2/1 ) — where the third arrow is the outer action determined by the p Π 2/1 exact sequence 1 → Π 2/1 → Π 2 → Π 1 → 1 — in a fashion compatible with the projection p Π {1,2}/{2} | Π 2/1 : Π 2/1  Π {2} and ∼ ∼ the given outer isomorphisms Π {2} → Π 1 → Π G . Proof. First, we verify assertion (i). The existence of such a (Π w  ) 2 ⊆ Π 2 follows immediately from the various definitions involved. Thus, it remains to verify the uniqueness of such a (Π w  ) 2 . Let (Π w  ) 2 ⊆ Π 2 be as in assertion (i) and γ ∈ Π 2 an element such that the conjugate (Π w  ) γ 2 of (Π w  ) 2 by γ satisfies the condition on “(Π w  ) 2 ” stated in assertion (i). Then since Π w  is commensurably terminal in Π 1 [cf. [CmbGC], Proposition 1.2, (ii)], it holds that the image of γ via p Π 2/1 is contained in Π w  . Thus — by multiplying γ by a suitable element of (Π w  ) 2 — we may assume without loss of generality that γ ∈ Π 2/1 . In particular, since Π v  new ⊆ (Π w  ) 2/1 ∩(Π w  ) γ 2/1 — where we write (Π w  ) γ 2/1 = Π 2/1 ∩(Π w  ) γ 2 — is not abelian [cf. [CmbGC], Remark 1.1.3], it follows immediately from [NodNon], Lemma 1.9, (i), that (Π w  ) 2/1 = (Π w  ) γ 2/1 . Thus, since (Π w  ) 2/1 is commensurably terminal in Π 2/1 [cf. [CmbGC], Proposition 1.2, (ii)], it holds that γ ∈ (Π w  ) 2/1 . This completes the proof of assertion (i). Assertions (ii), (iii), (iv) follow immediately from the various def- initions involved [cf. also [CmbGC], Propositions 1.2, (ii), and 1.5, (i), as well as the proofs of [CmbCsp], Proposition 1.5, (iii); [CbTpI], Proposition 2.11]. Finally, we verify assertion (v). It follows immediately from the def- inition of “Out FC ((Π (−) ) 2 ) G-node ” [cf. Definitions 4.6, (ii); 4.9] that, for each z  ∈ { v , w},  there exists a lifting α  z  ∈ Aut((Π z  ) 2 ) of α z  (G) such that  z  , then if we write ( α z  ) 1 for the automorphism of Π z  determined by α ( α z  ) 1 (Π e  ) = Π e  . Next, let us observe that it follows immediately from assertion (ii) that the automorphisms ( α v  ) 1 , ( α w  ) 1 [i.e., determined  w  ] determine an automorphism α  1 | Π v  , w  of Π v  , w  . by the liftings α  v  , α Moreover, let us also observe that it follows immediately from Theo- rem 4.2, (iii) [cf. also the definition of profinite Dehn multi-twists given in [CbTpI], Definition 4.4], that the assignment “α 1 → α 1 | Π v  , w  ” implicit in assertion (iv) is injective. Thus, one verifies immediately from the definition of profinite Dehn multi-twists that one may choose the re-  w  of α v , α w so that ( α v  ) 1 (Π e  ) = ( α w  ) 1 (Π e  ) = Π e  , spective liftings α  v  , α def 120 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI and, moreover, the outomorphism of Π v  , w  determined by the result- ing automorphism α  1 | Π v  , w  coincides with the outomorphism α 1 | Π v  , w  of assertion (iv). Now we claim that the following assertion holds: Claim 4.12.A: Write ( α z  ) 2/1 for the automorphism of  z  and (α z  ) 2/1 for the outo- (Π z  ) 2/1 determined by α α z  ) 2/1 . Then — morphism of (Π z  ) 2/1 determined by ( ∼ relative to the natural identification Π H z  ◦ → (Π z  ) 2/1 of assertion (iii) — it holds that ∈ Aut |Brch(H z ◦ )| (H z ◦ ) ∼ → Out((Π z  ) 2/1 )) . (⊆ Out(Π H z  ◦ ) Indeed, careful inspection of the various definitions involved reveals that Claim 4.12.A follows immediately from Theorem 4.7 [together with the commensurable terminality of the subgroup Π e  ⊆ Π z  — cf. [CmbGC], Proposition 1.2, (ii)]. Thus — by replacing α  z  by the com- posite of α  z  with an inner automorphism determined by conjugation by a suitable element of (Π z  ) 2/1 — we may assume without loss of generality that α  z  (Π e  z  ◦ ) = Π e  z  ◦ . Moreover, since [cf. Claim 4.12.A] α  z  preserves the (Π z  ) 2/1 -conjugacy classes of Π z  ◦ and Π v  new , and the ∼ verticial subgroups Π z  ◦ , Π v  new ⊆ Π G 2/1 ← Π 2/1 are the unique verti- ∼ cial subgroups of Π G 2/1 ← Π 2/1 associated to z  (G) ◦ , v new ∈ Vert(G 2/1 ), respectively, such that Π e  z  ◦ ⊆ Π z  ◦ , Π e  z  ◦ ⊆ Π v  new [cf. [CmbGC], Propo- sition 1.5, (i)], we thus conclude that α  z  (Π z  ◦ ) = Π z  ◦ , α  z  (Π v  new ) = Π v  new . Next, write (α z  ) z  ◦ , (α z  ) v  new for the respective outomorphisms of Π z  ◦ , Π v  new determined by α  z  . Now we claim that the following assertion holds: Claim 4.12.B: It holds that (α z  ) 2/1 (α v  ) v  new = (α w  ) v  new . Moreover, if v = w, i.e., G is cyclically primitive, then ∼ — relative to the natural outer isomorphism Π v  ◦ → Π w  ◦ [where we note that if v = w, then Π v  ◦ is a Π 2/1 - conjugate of Π w  ◦ ] — it holds that (α v  ) v  ◦ = (α w  ) w  ◦ . Indeed, the equality (α v  ) v  new = (α w  ) v  new follows from the definition of Glu(Π 2 ). Next, suppose that G is cyclically primitive. To verify the equality (α v  ) v  ◦ = (α w  ) w  ◦ , let us observe that, for each z  ∈ { v , w},  the p Π {1,2}/{2} ∼ composite Π z  ◦ → Π 2  Π {2} → Π G is injective [and its image is a verticial subgroup of Π G associated to z  (G) ∈ Vert(G)]. Thus, to verify the equality (α v  ) v  ◦ = (α w  ) w  ◦ , it suffices to verify that the outomor- phism of the image of Π v  ◦ in Π {2} induced by (α v  ) v  ◦ coincides with the outomorphism of the image of Π w  ◦ in Π {2} induced by (α w  ) w  ◦ . On the COMBINATORIAL ANABELIAN TOPICS II 121 other hand, this follows immediately from the fact that both α  v  and α  w  are liftings of the same outomorphism α v = α w of “(Π v ) 2 ”=“(Π w ) 2 ” [cf. [CmbCsp], Proposition 1.2, (iii)]. This completes the proof of Claim 4.12.B. Next, let us observe that it follows immediately from the various definitions involved that if G is noncyclically primitive (respectively, cyclically primitive), then #Vert((G 2/1 ) {e v ◦ } ) = 2 (respectively, = 1), and that, relative to the correspondence discussed in [CbTpI], Propo- sition 2.9, (i), (3), H v ◦ and G 2/1 | w ◦ (G) (respectively, H v ◦ ) correspond(s) to the two vertices (respectively, the unique vertex) of (G 2/1 ) {e v ◦ } . Next, let us observe the following equalities [cf. the notation of [CbTpI], Definition 3.8, (ii)]: χ H v ◦ ((α v  ) 2/1 ) = = = = χ H z ◦ | v new ((α v  ) v  new ) χ H v ◦ | v new ((α w  ) v  new ) χ H w ◦ ((α w  ) 2/1 ) χ G 2/1 | w ◦ (G) ((α w  ) w  ◦ ) [cf. [CbTpI], Corollary 3.9, (iv)] [cf. Claim 4.12.B] [cf. [CbTpI], Corollary 3.9, (iv)] [cf. [CbTpI], Corollary 3.9, (iv)]. Now it follows immediately from these equalities, together with Claim 4.12.A, that the data ((α v  ) 2/1 , (α w  ) w  ◦ ) ∈ Aut(H v ◦ ) × Aut(G 2/1 | w ◦ (G) ) (respectively, (α v  ) 2/1 ∈ Aut(H v ◦ ) ) may be regarded as an element of Glu brch ((G 2/1 ) {e v ◦ } ) [cf. Defini- tion 4.1, (iii)]. Thus, by applying the exact sequence of Theorem 4.2, (iii) [cf. also Remark 4.9.1], we conclude that there exists an element v ] ∈ Aut |Brch((G 2/1 ) {ev ◦ } )| ((G 2/1 ) {e v ◦ } ) α 2/1 [ of a collection of outomorphisms of Φ (G 2/1 ) {ev ◦ } ∼ −→ Π (G 2/1 ) {e ◦ } v ∼ Π G 2/1 −→ Π 2/1 [i.e., contained in the image of Aut((G 2/1 ) {e v ◦ } ) → Out(Π 2/1 ) — cf. [CbTpI], Definition 2.10] that admits a natural structure of torsor over Dehn((G 2/1 ) {e v ◦ } ) (⊆ Aut |Brch((G 2/1 ) {ev ◦ } )| ((G 2/1 ) {e v ◦ } )). A similar argument yields the existence of an element  ∈ Aut |Brch((G 2/1 ) {ew ◦ } )| ((G 2/1 ) {e w ◦ } ) α 2/1 [ w] of a collection of outomorphisms of Φ (G Π (G 2/1 ) {e ◦ } w 2/1 ) {ew ◦ } ∼ −→ ∼ Π G 2/1 −→ Π 2/1 [i.e., contained in the image of Aut((G 2/1 ) {e w ◦ } ) → Out(Π 2/1 )] that admits a natural structure of torsor over Dehn((G 2/1 ) {e w ◦ } ) (⊆ Aut |Brch((G 2/1 ) {ew ◦ } )| ((G 2/1 ) {e w ◦ } )). 122 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Now we claim that the following assertion holds: Claim 4.12.C: For each z  ∈ { v , w},  the automorphism ( α z  ) 1 of Π z  is compatible with the outomorphism α 2/1 [ z ] of Π 2/1 relative to the homomorphism Π z  → Π 1 → Out(Π 2/1 ) — where the second arrow is the natural outer action determined by the exact sequence p Π 2/1 1 −→ Π 2/1 −→ Π 2 −→ Π 1 −→ 1 . v ], it follows im- Indeed, to verify the compatibility of ( α v  ) 1 and α 2/1 [ mediately from the various definitions involved that it suffices to verify def α v  ) 1 (σ) ∈ Π v  , then there exist that, for each σ ∈ Π v  , if we write τ = ( liftings σ  , τ  ∈ Π 2 of σ, τ ∈ Π v  , respectively, such that the equality [which is in fact independent of the choice of liftings] v ] ◦ [Inn( σ )] ◦ α 2/1 [ v ] −1 = [Inn( τ )] ∈ Out(Π 2/1 ) α 2/1 [ — where we write “Inn(−)” for the automorphism of Π 2/1 determined by conjugation by “(−)” and “[Inn(−)]” for the outomorphism of Π 2/1 determined by this automorphism — holds. To this end, let σ  ∈ (Π v  ) 2 be a lifting of σ ∈ Π v  . Then since (Π v  ) 2/1 ⊆ (Π v  ) 2 is normal, Inn( σ ) preserves (Π v  ) 2/1 . Next, let us observe that the semi-graph of anabelioids structure of (G 2/1 ) {e v ◦ } [with respect to which w ◦ is a vertex if G is noncyclically primitive and, moreover, with respect to which e w ◦ is a node in both the cyclically primitive and noncyclically primitive cases] may be thought of as the semi-graph of anabelioids structure on the fiber subgroup Π 2/1 [cf. Definition 3.1, (iii)] arising from a point of X log that lies in the 1-interior of the irreducible component of X log corresponding to v. Now it follows immediately from this observation that Inn( σ ) preserves the Π 2/1 -conjugacy class of Π w  ◦ , as well as the Π 2/1 -conjugacy class of Π e  w  ◦ = (Π v  ) 2/1 ∩ Π w  ◦ if G is noncyclically primitive (respectively, pre- serves the Π 2/1 -conjugacy class of Π e  w  ◦ if G is cyclically primitive). By considering the various Π 2/1 -conjugates of Π e  w  ◦ and Π w  ◦ and applying [CmbGC], Propositions 1.2, (ii); 1.5, (i), we thus conclude that Inn( σ ) preserves the (Π v  ) 2/1 -conjugacy classes of Π e  w  ◦ , Π w  ◦ if G is noncycli- cally primitive (respectively, preserves the (Π v  ) 2/1 -conjugacy class of Π e  w  ◦ if G is cyclically primitive). In particular — by multiplying σ  by a suitable element of (Π v  ) 2/1 — we may assume without loss of gener- ality that Inn( σ ) preserves (Π v  ) 2/1 , Π w  ◦ , and Π e  w  ◦ in the noncyclically primitive case (respectively, preserves (Π v  ) 2/1 and Π e  w  ◦ in the cyclically primitive case). Next, let us observe that one verifies easily [cf. Lemma 3.6, (iv)] that p Π {1,2}/{2} the composite Π e  w  ◦ → Π 2  Π {2} surjects onto a nodal subgroup ∼ of Π G ← Π {2} associated to e ∈ Node(G). Thus, since Inn( σ ) preserves Π e  w  ◦ , it follows [cf. [CmbGC], Proposition 1.2, (ii)] that the image of COMBINATORIAL ANABELIAN TOPICS II σ  ∈ Π 2 via Π 2 p Π {1,2}/{2}  123 Π {2} is contained in the image of the composite p Π {1,2}/{2} Π e  w  ◦ → Π 2  Π {2} . In particular — by multiplying σ  by a suitable element of Π e  w  ◦ (⊆ (Π v  ) 2/1 ) — we may assume without loss of generality that σ  ∈ Ker(p Π {1,2}/{2} ). A similar argument implies that α v  ) 1 (σ) ∈ Π v  such that Inn( τ ) there exists a lifting τ  ∈ (Π v  ) 2 of τ = ( preserves (Π v  ) 2/1 , Π w  ◦ , Π e  w  ◦ if G is noncyclically primitive (respectively, preserves (Π v  ) 2/1 and Π e  w  ◦ if G is cyclically primitive), and, moreover, τ  ∈ Ker(p Π {1,2}/{2} ). α v  ) 1 of (Π v  ) 2/1 , Π v  , respec- Now since the automorphisms ( α v  ) 2/1 , ( tively, arise from the automorphism α  v  of (Π v  ) 2 , it follows immediately v ] that the equality from the construction of α 2/1 [ v ] ◦ [Inn( σ )] ◦ α 2/1 [ v ] −1 = [Inn( τ )] α 2/1 [ holds upon restriction to [an equality of outomorphisms of] (Π v  ) 2/1 . Moreover, if G is noncyclically primitive, then since the composite p Π {1,2}/{2} Π w  ◦ → Π 2  Π {2} is injective [and its image is a verticial sub- ∼ group of Π G ← Π {2} associated to w ∈ Vert(G) — cf. Lemma 3.6, (iv)], to verify the restriction of the equality v ] ◦ [Inn( σ )] ◦ α 2/1 [ v ] −1 = [Inn( τ )] α 2/1 [ to [an equality of outomorphisms of] Π w  ◦ , it suffices to verify that the outomorphism of the image of Π w  ◦ in Π {2} induced by the product v ] ◦ [Inn( σ )] ◦ α 2/1 [ v ] −1 ◦ [Inn( τ )] −1 α 2/1 [ is trivial. On the other hand, this follows immediately from the fact that σ  , τ  ∈ Ker(p Π {1,2}/{2} ). Thus, in summary, the restriction of the equality in question [i.e., in the discussion immediately following Claim 4.12.C] to [an equality of outomorphisms of] (Π v  ) 2/1 holds. Moreover, if G is noncyclically prim- itive, then the restriction of the equality in question to [an equality of outomorphisms of] Π w  ◦ holds. In particular, it follows immediately from the displayed exact sequence of Theorem 4.2, (iii) [cf. also Re- mark 4.9.1], that the product α 2/1 [ v ] ◦ [Inn( σ )] ◦ α 2/1 [ v ] −1 ◦ [Inn( τ )] −1 is contained in Dehn((G 2/1 ) {e v ◦ } ). Thus — by considering the outo- morphism of Π {2} induced by the above product — one verifies eas- ily from [CbTpI], Theorem 4.8, (iv), together with the fact that σ  , Π τ  ∈ Ker(p {1,2}/{2} ), that the equality in question holds. This completes v ]. The compatibility of the proof of the compatibility of ( α v  ) 1 and α 2/1 [  follows from a similar argument. This completes the ( α w  ) 1 and α 2/1 [ w] proof of Claim 4.12.C. Next, we claim that the following assertion holds: 124 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Claim 4.12.D: The difference α 2/1 [ v ]◦α 2/1 [ w]  −1 ∈ Out(Π 2/1 ) ∼ is contained in Dehn(G 2/1 ) (⊆ Out(Π G 2/1 ) ← Out(Π 2/1 )). Indeed, this follows immediately from the two displayed equalities of Claim 4.12.B, together with the construction of α 2/1 [ v ], α 2/1 [ w].  This completes the proof of Claim 4.12.D. Thus, it follows immediately from Claim 4.12.D, together with the existence of the natural isomorphism ∼ Dehn((G 2/1 ) {e v ◦ } ) ⊕ Dehn((G 2/1 ) {e w ◦ } ) −→ Dehn(G 2/1 ) [cf. [CbTpI], Theorem 4.8, (ii), (iv)], that — by replacing α 2/1 [ v ], α 2/1 [ w]  by the composites of α 2/1 [ v ], α 2/1 [ w]  with suitable elements of Dehn((G 2/1 ) {e v ◦ } ), Dehn((G 2/1 ) {e w ◦ } ), respectively [where we re- v ], α 2/1 [ w]  belong to torsors over call that the outomorphisms α 2/1 [ Dehn((G 2/1 ) {e v ◦ } ), Dehn((G 2/1 ) {e w ◦ } ), respectively] — we may as- sume without loss of generality that α 2/1 [ v ] = α 2/1 [ w]  . def v ] = α 2/1 [ w].  Then it follows immediately from Write β 2/1 = α 2/1 [ Claim 4.12.C, together with the fact that Π v  , w  is topologically gener- ated by Π v  , Π w  ⊆ Π v  , w  [cf. assertion (ii)], that the outomorphism β 2/1 of Π 2/1 is compatible with the automorphism α  1 | Π v  , w  of Π v  , w  [i.e., the α w  ) 1 — cf. the discussion immedi- automorphism induced by ( α v  ) 1 , ( ately preceding Claim 4.12.A], relative to the composite Π v  , w  → Π 1 → Out(Π 2/1 ), where the second arrow is the outer action determined by the displayed exact sequence of Claim 4.12.C. In particular, by con- ∼ out sidering the natural isomorphism Π 2 | Π v  , w  → Π 2/1  Π v  , w  [cf. the discussion entitled “Topological groups” in [CbTpI], §0], we obtain an outomorphism β v  , w  of Π 2 | Π v  , w  which, by construction, satisfies the four conditions listed in assertion (v). This completes the proof of assertion (v).  Lemma 4.13 (Glueability of combinatorial cuspidalizations in the case of precisely one node). Suppose that n = 2, and that [cf. Definition 4.11] is surjective. #Node(G) = 1. Then ρ brch 2 Proof. If G is noncyclically primitive [cf. [CbTpI], Definition 4.1], then the surjectivity of ρ brch follows immediately from Lemma 4.12, (v) [cf. 2 also [CmbCsp], Proposition 1.2, (i)], together with the fact that the natural injection Π v  , w  → Π 1 is an isomorphism [cf. Lemma 4.12, (ii)]. in the case where Thus, it remains to verify the surjectivity of ρ brch 2 G is cyclically primitive [cf. [CbTpI], Definition 4.1]. Since we are in the situation of [CbTpI], Lemma 4.3, we shall apply the notational conventions established in [CbTpI], Lemma 4.3. Also, we shall write COMBINATORIAL ANABELIAN TOPICS II 125 Vert(G) = {v}, Node(G) = {e}. Let x ∈ X 2 (k) be a k-rational geomet- ric point of X 2 such that x {1} ∈ X(k) [cf. Definition 3.1, (i)] lies on the unique node of X log [i.e., which corresponds to e]. Recall from [CbTpI], Lemma 4.3, (i), that we have a natural exact sequence 1 −→ π 1 temp (G ∞ ) −→ π 1 temp (G) −→ π 1 top (G) −→ 1 .  ∞ ∈ π 1 temp (G) Let γ ∞ ∈ π 1 top (G) be a generator of π 1 top (G) (≃ Z) and γ ∼ a lifting of γ ∞ . By abuse of notation, write γ  ∞ ∈ Π G ← Π 1 for the ∼ image of γ  ∞ ∈ π 1 temp (G) via the natural injection π 1 temp (G) → Π G ← Π 1 [cf. the evident pro-Σ generalization of [SemiAn], Proposition 3.6, (iii); [RZ], Proposition 3.3.15]. Next, let us fix a verticial subgroup temp (G ∞ ) ⊆) π 1 temp (G) Π temp v  (0) ⊆ (π 1  that lifts the of π 1 temp (G) that corresponds to a vertex v  (0) ∈ Vert( G) vertex V (0) ∈ Vert(G ∞ ) [cf. [CbTpI], Lemma 4.3, (iii)]. Thus, for each a  ∞ , we obtain a integer a ∈ Z, by forming the conjugate of Π temp v  (0) by γ verticial subgroup temp (G ∞ ) ⊆) π 1 temp (G) Π temp v  (a) ⊆ (π 1  that lifts the of π 1 temp (G) associated to some vertex v  (a) ∈ Vert( G) vertex V (a) ∈ Vert(G ∞ ) [cf. [CbTpI], Lemma 4.3, (iii), (vi)]. Write Π v  (a) ⊆ Π G temp for the image of Π temp (G) in Π G . v  (a) ⊆ π 1 Next, let us suppose that γ  ∞ was chosen in such a way that, for each a ∈ Z, the intersection N ( v (a)) ∩ N ( v (a + 1)) consists of a unique node  n  (a, a + 1) ∈ Node( G) that lifts the node N (a + 1) ∈ Node(G ∞ ) [cf. [CbTpI], Lemma 4.3, (iii)]. [One verifies easily that such a γ  ∞ always exists.] Then let us observe that, for each a ≤ b ∈ Z, we have a nat- ural morphism of semi-graphs of anabelioids G [a,b] → G ∞ [cf. [CbTpI], Lemma 4.3, (iv)], which induces injections [cf. the evident pro-Σ gen- eralizations of [SemiAn], Example 2.10; [SemiAn], Proposition 2.5, (i); [SemiAn], Proposition 3.6, (iii); [RZ], Proposition 3.3.15] π 1 temp (G [a,b] ) → π 1 temp (G ∞ ), Π G [a,b] → Π G — where we write, respectively, π 1 temp (G [a,b] ), Π G [a,b] for the tempered, pro-Σ fundamental groups of the semi-graph of anabelioids G [a,b] of pro-Σ PSC-type — which are well-defined up to composition with in- ner automorphisms. By choosing appropriate basepoints [cf. also our choice of γ  ∞ ], these inner automorphism indeterminacies may be elimi- nated in such a way that, for each a ≤ c ≤ b, the resulting injections are compatible with one another and, moreover, their images contain the ∼ temp (G ∞ ), Π v  (c) ⊆ Π G ← Π 1 , respectively. Then, subgroups Π temp v  (c) ⊆ π 1 126 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI relative to the resulting inclusions, Π temp  (c) form verticial subgroups v  (c) , Π v temp of π 1 (G [a,b] ), Π G [a,b] associated to the vertex of G [a,b] corresponding to V (c) [cf. [CbTpI], Lemma 4.3, (iii)]. In particular, we have a natural isomorphism ∼ def Π [a,a+1] = Π v  (a), v (a+1) −→ Π G [a,a+1] [cf. Lemma 4.12, (ii)]. Let us write def Π 2 | [a,a+1] = Π 2 | Π [a,a+1] ⊆ Π 2 [cf. Lemma 4.12, (ii)]; def Π [a] = Π v  (a) ; def Π 2 | [a] = Π 2 × Π 1 Π [a] ⊆ Π 2 | [a−1,a] , Π 2 | [a,a+1] . Next, we claim that the following assertion holds: Claim 4.13.A: The profinite group Π G is topologically generated by Π [0] ⊆ Π G and γ  ∞ ∈ Π G . Indeed, let us first observe that it follows immediately from a similar argument to the argument applied in the proof of [CmbCsp], Propo- sition 1.5, (iii) [i.e., in essence, from the “van Kampen Theorem” in elementary algebraic topology], that • the image of the natural homomorphism lim π 1 temp (G [−a,a] ) −→ π 1 temp (G ∞ ) −→ a≥0 — where the inductive limit is taken in the category of tem- pered groups [cf. [SemiAn], Definition 3.1, (i); [SemiAn], Ex- ample 2.10; [SemiAn], Proposition 3.6, (i)] — is dense; • for each nonnegative integer a, the tempered group π 1 temp (G [−a,a] ) [cf. [SemiAn], Example 2.10; [SemiAn], Proposition 3.6, (i)] is temp temp (G [−a,a] ). topologically generated by Π temp v  (−a) , . . . , Π v  (a) ⊆ π 1 In particular, it follows immediately from the exact sequence of [CbTpI], Lemma 4.3, (i), that the tempered group π 1 temp (G) [cf. [SemiAn], Ex- ample 2.10; [SemiAn], Proposition 3.6, (i)] is topologically generated temp (G) and γ  ∞ ∈ π 1 temp (G). Thus, Claim 4.13.A fol- by Π temp v  (0) ⊆ π 1 lows immediately from the fact that the image of the natural injection π 1 temp (G) → Π G is dense. This completes the proof of Claim 4.13.A. For a ∈ Z, let us write [a,a+1] def G 2/1 = G 2∈{1,2},x [cf. Definition 3.1, (iii)], where we fix isomorphisms ∼ Π 2/1 −→ Π G [a,a+1] , 2/1 ∼ Π {2} −→ Π G 2∈{2},x = Π G [the latter of which is to be understood as being independent of a ∈ Z] as in [i.e., that belong to the collections of isomorphisms that constitute COMBINATORIAL ANABELIAN TOPICS II 127 the outer isomorphisms of the final display of] Definition 3.1, (iii), to be isomorphisms [cf. the discussion of the final portion of Lemma 4.12, [a,a+1] (v)] such that the semi-graph of anabelioids structure on G 2/1 is the semi-graph of anabelioids structure determined by the resulting composite ∼ ∼ Π n  (a,a+1) → Π G ← Π 1 → Out(Π 2/1 ) → Out(Π G [a,a+1] ) 2/1 — where we write Π n  (a,a+1) ⊆ Π G for the nodal subgroup of Π G as- sociated to the unique element n  (a, a + 1) ∈ N ( v (a)) ∩ N ( v (a + 1)), and the third arrow arises from the outer action determined by the p Π 2/1 exact sequence 1 → Π 2/1 → Π 2 → Π 1 → 1 — in a fashion compatible with the projection p Π {1,2}/{2} | Π 2/1 : Π 2/1  Π {2} and the isomorphisms ∼ ∼ Π {2} → Π G ← Π 1 [cf. Definition 3.1, (ii)]. Here, we note that, for a, ∼ [a,a+1] ∼ [b,b+1] → G 2∈{1,2},x → G 2/1 b ∈ Z, there exist natural isomorphisms G 2/1 of semi-graphs of anabelioids of pro-Σ PSC-type [induced by conjuga- b−a tion by γ  ∞ ]. On the other hand, it is not difficult to show [although we shall not use this fact in the present proof!] that the well-known injectivity of the homomorphism Π 1 → Out(Π 2/1 ) of the above display [cf. [CbTpI], Lemma 5.4, (i), (ii), (iii); [CbTpI], Theorem 4.8, (iv); [Asd], Theorem 1; [Asd], the Remark following the proof of Theorem 1; [CmbGC], Proposition 1.2, (i), (ii)] implies that when a  = b, the composite ∼ ∼ Π G [a,a+1] ← Π 2/1 → Π G [b,b+1] 2/1 2/1 in fact fails to be graphic! For each a ∈ Z, let us write [a,a+1][a] def G 2/1 [a,a+1] = (G 2/1 [a,a+1][a+1] def ) {e v(a) ◦ } , G 2/1 [a,a+1] = (G 2/1 ) {e v(a+1) ◦ } — where we write e v(a) ◦ , e v(a+1) ◦ for the nodes “e z ◦ ” of Lemma 4.12, (iii), that occur, respectively, in the cases where the pair “(G 2/1 , z  ◦ )” [a,a+1] [a,a+1] is taken to be (G 2/1 , v  (a) ◦ ); (G 2/1 , v  (a + 1) ◦ ). Then one verifies easily [cf. Lemma 4.12, (i), (iii)] that the composite ∼ ∼ ∼ ∼ Π G [a−1,a][a] → Π G [a−1,a] ← Π 2/1 → Π G [a,a+1] ← Π G [a,a+1][a] 2/1 2/1 2/1 2/1 — where the first and fourth arrows are the natural specialization outer isomorphisms [cf. [CbTpI], Definition 2.10], and the second and third arrows are the isomorphisms fixed above — is graphic. In light of this observation, it makes sense to write [a] def [a−1,a][a] ∼ G 2/1 = G 2/1 [a,a+1][a] → G 2/1 [cf. Figure 4 below]. This notation allows us to express the graphicity observed above in the following way: 128 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI The composites ∼ ∼ ∼ Π [a] → Π G ← Π 1 → Out(Π 2/1 ) → Out(Π G [a−1,a] ) ← Out(Π G [a] ), 2/1 2/1 ∼ ∼ ∼ Π [a] → Π G ← Π 1 → Out(Π 2/1 ) → Out(Π G [a,a+1] ) ← Out(Π G [a] ) 2/1 2/1 — where the third arrows in each line of the display arise from the outer action determined by the exact p Π 2/1 sequence 1 → Π 2/1 → Π 2 → Π 1 → 1, the fourth arrows are the isomorphisms induced by the isomor- ∼ ∼ phisms Π 2/1 → Π G [a−1,a] and Π 2/1 → Π G [a,a+1] fixed 2/1 2/1 above, and the fifth arrows are the isomorphisms in- duced by the natural specialization outer isomorphisms [cf. [CbTpI], Definition 2.10] — factor through [a] Aut(G 2/1 ) ⊆ Out(Π G [a] ). 2/1 [a−1,a] [a] G 2/1 • •  ·· ·· ·· ·· ·· · G 2/1  • • ·· ·· ·· ·· ·· · ∨ ∨ v  (a) v  (a + 1) •    Π [a−1,a] [a−1,a] Figure 4: G 2/1 • ·· ·· ·· ·· ·· · ∨ v  (a − 1) ··· •  [a,a+1] G 2/1  Π [a,a+1] [a] •  ··· [a,a+1] , G 2/1 , and G 2/1 Now we turn to the verification of the surjectivity of the homomor- phism ρ brch . Let α v ∈ Glu(Π 2 ) (⊆ Out FC ((Π v ) 2 ) G-node ). Write (α v ) 1 ∈ 2 Glu(Π 1 ) for the image of α v ∈ Glu(Π 2 ) via the injection of Lemma 4.10, (i). Let α 1 ∈ Aut |Brch(G)| (G) be such that ρ brch (α 1 ) = (α v ) 1 ∈ Glu(Π 1 ) 1 COMBINATORIAL ANABELIAN TOPICS II 129 [cf. Theorem 4.2, (iii); Definition 4.11]. Now, by applying Lemma 4.12, (v), in the case where we take the pair “( v , w)”  to be ( v (0), v  (1)), we def obtain an outomorphism β [0,1] = β v  (0), v (1) [α 1 ] [cf. Lemma 4.12, (v)] of Π 2 | [0,1] [cf. the notation of the discussion preceding Claim 4.13.A]. Let † • β  [0,1] ∈ Aut(Π 2 | [0,1] ) be an automorphism that lifts β [0,1] ∈ Out(Π 2 | [0,1] ) and preserves the subgroup Π n  (0,1) ⊆ Π [0,1] [cf. condition (4) of Lemma 4.12, (v)] and   ∞ ∈ Π 1 . • γ  ∞ ∈ Π 2 a lifting of γ Then since [as is easily verified] Π 2 | [1,2] [cf. the notation of the dis-  cussion preceding Claim 4.13.A] is the conjugate of Π 2 | [0,1] by γ  ∞ , by †  conjugating β  [0,1] by the inner automorphism determined by γ  ∞ , we obtain an automorphism β  † of Π 2 | [1,2] , whose associated outomor- [1,2] phism we denote by β [1,2] . Now we claim that the following assertion holds: Claim 4.13.B: There exist automorphisms β  [0,1] , β  [1,2] of Π 2 | [0,1] , Π 2 | [1,2] that lift β [0,1] , β [1,2] , respectively, such that (i) the outomorphisms of Π 2/1 (⊆ Π 2 | [0,1] , Π 2 | [1,2] ) de- termined by β  [0,1] , β  [1,2] coincide; (ii) the automorphism of Π [0,1] determined by the au- tomorphism β  [0,1] preserves the subgroups Π n  (0,1) , Π [0] , Π [1] ⊆ Π [0,1] ; † (iii) β  [0,1] = β  [0,1] , and β  [1,2] is the post-composite of β  † with an inner automorphism arising from an [1,2] element of Π 2 | [1] . † Indeed, observe that there exist automorphisms β  [0,1] , β  [1,2] [e.g., β  [0,1] , † β  ] of Π 2 | [0,1] , Π 2 | [1,2] that lift β [0,1] , β [1,2] , respectively, such that [1,2] • the outomorphisms ( β  [0,1] ) 2/1 , ( β  [1,2] ) 2/1 of Π 2/1 determined by β  [0,1] , β  [1,2] are contained in [0,1] Aut |Brch(G 2/1 )| (G 2/1 ), [0,1] [1,2] Aut |Brch(G 2/1 )| (G 2/1 ) (⊆ Out(Π 2/1 )), [1,2] respectively, and, • conditions (ii), (iii) of Claim 4.13.B are satisfied [cf. the discussion of the final portion of Lemma 4.12, (v); Lemma 4.12, (v), (1); [CmbGC], Proposition 1.5, (i)]. In particular, it follows that, ∼ relative to the specialization outer isomorphisms Π G [1] → Π G [0,1] , Π G [1] 2/1 ∼ 2/1 2/1 → Π G [1,2] that appeared in the discussion following the proof of Claim 2/1 4.13.A, together with the natural inclusion of [CbTpI], Proposition 2.9, 130 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (ii), [1] ( β  [0,1] ) 2/1 , ( β  [1,2] ) 2/1 ∈ Aut |Brch(G 2/1 )| (G 2/1 ) (⊆ Out(Π 2/1 )) . [1] Moreover, it follows immediately from condition (3) of Lemma 4.12, (v), applied in the case of β [0,1] , together with the definition of β [1,2] , that the outomorphisms of the configuration space subgroup     Π 2 ⊇ Π 2 | [0,1] ⊇ ⊆ Π 2 | [1,2] ⊆ Π 2 (Π v  (1) ) 2 associated to the vertex v  (1) determined by β [0,1] , β [1,2] coincide with α v . Now let us recall from the above discussion that the composite ∼ Π [1] → Π 1 → Out(Π 2/1 ) → Out(Π G [1] ) 2/1 factors through [1] Aut(G 2/1 ) ⊆ Out(Π G [1] ) . 2/1 Thus, it follows immediately from the displayed exact sequence of The- orem 4.2, (iii) [cf. also Remark 4.9.1], that — after possibly replacing β  [1,2] by the post-composite of β  [1,2] with an inner automorphism arising from a suitable element of Π 2 | [1] [which does not affect the validity of conditions (ii), (iii) of Claim 4.13.B] — if we write [1] def [1] |Brch(G 2/1 )| δ = ( β  [0,1] ) 2/1 ◦ ( β  [1,2] ) −1 (G 2/1 ) (⊆ Out(Π 2/1 )) , 2/1 ∈ Aut [1] then it holds that δ ∈ Dehn(G 2/1 ). Next, let us observe that, for a ∈ {0, 1}, since β  [a,a+1] preserves the Π 2/1 -conjugacy class of cuspidal inertia subgroups associated to the di- agonal cusp [cf. condition (3) of Lemma 4.12, (v)], it follows from a similar argument to the argument applied in the proof of [CmbCsp], Proposition 1.2, (iii), that the outomorphism ( β  [a,a+1] ) {2} of Π {2} in- duced by β  [a,a+1] on the quotient ∼ Π G [1] ← Π 2/1 → Π 2 2/1 p Π {1,2}/{2}  Π {2} ∼ is compatible, relative to the natural inclusion Π [a,a+1] → Π 1 → Π {2} , with the outomorphism α 1 | Π [a,a+1] [cf. condition (4) of Lemma 4.12, (v)]. Since an element of Aut |Brch(G)| (G) is completely determined by its restriction to Aut(G [a,a+1] ) [cf. [CbTpI], Definition 4.4; [CbTpI], Remark 4.8.1], we thus conclude that, relative to the natural outer ∼ ∼ isomorphisms Π {2} → Π 1 → Π G , it holds that ( β  [a,a+1] ) {2} = α 1 . In particular, it follows that the element of Aut |Brch(G)| (G) induced by δ ∈ Aut [1] |Brch(G 2/1 )| [1] (G 2/1 ) on the quotient Π G [1] 2/1 ∼ ← Π 2/1 → Π 2 p Π {1,2}/{2}  COMBINATORIAL ANABELIAN TOPICS II 131 ∼ Π {2} → Π G is trivial. On the other hand, let us observe that one verifies easily from [CbTpI], Theorem 4.8, (iii), (iv), that this composite ∼ Π G [1] ← Π 2/1 → Π 2 p Π {1,2}/{2}  2/1 ∼ Π {2} → Π G determines an isomorphism [1] ∼ Dehn(G 2/1 ) −→ Dehn(G) . Thus, we conclude that δ is the identity outomorphism of Π 2/1 . In particular, condition (i) of Claim 4.13.B is satisfied. This completes the proof of Claim 4.13.B. Next, let us fix an automorphism α  1 ∈ Aut(Π 1 ) that lifts α 1 ∈ ∼ Aut |Brch(G)| (G) ⊆ Out(Π G ) ← Out(Π 1 ) and preserves the subgroup Π n  (0,1) ⊆ Π 1 [hence also the subgroups Π [0] , Π [1] , Π [0,1] ⊆ Π 1 ], and whose restriction to Π [0,1] ⊆ Π 1 coincides with the automorphism of Π [0,1] de- termined by the automorphism β  [0,1] of Π 2 | [0,1] . [One verifies easily that such an α  1 always exists — cf. Lemma 4.12, (v), (4); Claim 4.13.B, (ii).] Write β 2/1 ∈ Out(Π 2/1 ) for the outomorphism of Π 2/1 ⊆ Π 2 | [0,1] determined by β  [0,1] [or, equivalently, β  [1,2] — cf. Claim 4.13.B, (i)]. Now we claim that the following assertion holds: Claim 4.13.C: Write ρ : Π 1 → Out(Π 2/1 ) for the ho- momorphism determined by the exact sequence 1 → p Π 2/1 Π 2/1 → Π 2 → Π 1 → 1. Then −1 ρ( α 1 ( γ ∞ )) = β 2/1 ◦ ρ( γ ∞ ) ◦ β 2/1 ∈ Out(Π 2/1 ) . Indeed, let us first observe that it follows from conditions (i) and (iii) † of Claim 4.13.B, together with the definition of β  [1,2] , that there exists an element ∈ Π [1] such that −1 −1 γ ∞ ) ◦ β 2/1 = ρ( −1 ) ρ( γ ∞ ) ◦ β 2/1 ◦ ρ( (∗ 1 ) . Next, let us observe that if we write −1 η = α  1 ( γ ∞ ) · γ  ∞ ∈ Π 1 def (∗ 2 ), then it follows immediately from the commensurable terminality of Π [1] in Π 1 [cf. [CmbGC], Proposition 1.2, (ii)], together with our choices of  ∞ — which imply that α  1 and γ −1 α  1 ( γ ∞ ) · γ  ∞ · Π [1] · γ  ∞ · α  1 ( γ ∞ ) −1 = = = = = α  1 ( γ ∞ ) · Π [0] · α  1 ( γ ∞ ) −1 α  1 ( γ ∞ ) · α  1 (Π [0] ) · α  1 ( γ ∞ ) −1 −1 α  1 ( γ ∞ · Π [0] · γ  ∞ ) α  1 (Π [1] ) Π [1] — that η ∈ Π [1] . Thus, to verify Claim 4.13.C, it suffices to verify that ρ( ) = ρ(η). 132 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI To this end, let ζ ∈ Π [0] . Then, by our choice of γ  ∞ , it follows that −1  ∞ ∈ Π [1] . In particular, since the outomorphism β 2/1 arises γ  ∞ · ζ · γ from an automorphism β  [0,1] of Π 2 | [0,1] , which is an automorphism over the restriction of α  1 to Π [0,1] , it follows immediately that α 1 (ζ)) ◦ β 2/1 β 2/1 ◦ ρ(ζ) = ρ( (∗ 3 ) . −1 −1 β 2/1 ◦ ρ( γ ∞ · ζ · γ  ∞ ) = ρ( α 1 ( γ ∞ · ζ · γ  ∞ )) ◦ β 2/1 (∗ 4 ) . Thus, if we write −1 Θ  = ρ( · γ  ∞ · α  1 (ζ) · γ  ∞ · −1 ) ◦ β 2/1 ∈ Out(Π 2/1 ), def −1  ∞ · α  1 (ζ) · γ  ∞ · η −1 ) ◦ β 2/1 ∈ Out(Π 2/1 ), Θ η = ρ(η · γ def then Θ  = = = = = −1 ρ( · γ  ∞ · α  1 (ζ)) ◦ β 2/1 ◦ ρ( γ ∞ ) −1  ∞ ) ρ( · γ  ∞ ) ◦ β 2/1 ◦ ρ(ζ · γ −1 β 2/1 ◦ ρ( γ ∞ · ζ · γ  ∞ ) −1 γ ∞ · ζ · γ  ∞ )) ◦ β 2/1 ρ( α 1 ( Θ η [cf. (∗ 1 )] [cf. (∗ 3 )] [cf. (∗ 1 )] [cf. (∗ 4 )] [cf. (∗ 2 )] −1 — which thus implies that ρ(η −1 · ) commutes with ρ( γ ∞ · α 1 (ζ)· γ ∞ ). In −1 −1 particular, since γ  ∞ · α  1 (Π [0] )· γ ∞ = γ  ∞ ·Π [0] · γ ∞ = Π [1] , by allowing “ζ” to vary among the elements of Π [0] , it follows that ρ(η −1 · ) centralizes ρ(Π [1] ). On the other hand, it follows from [Asd], Theorem 1; [Asd], the Remark following the proof of Theorem 1, that ρ is injective. Thus, since , η ∈ Π [1] , we conclude that η −1 · ∈ Z(Π [1] ) = {1} [cf. [CmbGC], Remark 1.1.3]. This completes the proof of Claim 4.13.C. Now let us recall that the outomorphism β 2/1 of Π 2/1 of Claim 4.13.C arises from an automorphism β  [0,1] of Π 2 | [0,1] . Thus, it follows immedi- ately from Claims 4.13.A, 4.13.C that the outomorphism β 2/1 of Π 2/1 is compatible with the automorphism α  1 ∈ Aut(Π 1 ) relative to the homomorphism Π 1 → Out(Π 2/1 ) determined by the exact sequence p Π 2/1 1 → Π 2/1 → Π 2 → Π 1 → 1. In particular — by considering the ∼ out natural isomorphism Π 2 → Π 2/1  Π 1 [cf. the discussion entitled “Topological groups” in [CbTpI], §0] — we conclude that the outo- morphism β 2/1 ∈ Out(Π 2/1 ) extends to an outomorphism α 2 of Π 2 . On the other hand, it follows immediately from the various defini- (α 2 ) = α v ∈ Glu(Π 2 ) [cf. condition (3) of tions involved that ρ brch 2 Lemma 4.12, (v)], and that α 2 ∈ Out FC (Π 2 ) brch [cf. condition (2) of Lemma 4.12, (v); [CmbCsp], Proposition 1.2, (i)]. This completes the proof of Lemma 4.13 in the case where G is cyclically primitive, hence also of Lemma 4.13.  COMBINATORIAL ANABELIAN TOPICS II 133 Theorem 4.14 (Glueability of combinatorial cuspidalizations). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; n a positive integer; Σ a set of prime numbers which is either equal to the set of all prime numbers or of cardinality one; k an algebraically closed field of characteristic ∈ Σ; (Spec k) log the log scheme obtained by equipping Spec k with the log structure determined by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over (Spec k) log . Write G for the semi-graph of anabelioids of pro- Σ PSC-type determined by the stable log curve X log . For each positive integer i, write X i log for the i-th log configuration space of the stable log curve X log [cf. the discussion entitled “Curves” in “Notations and Conventions”]; Π i for the maximal pro-Σ quotient of the kernel of the natural surjection π 1 (X i log )  π 1 ((Spec k) log ). Then the following hold: (i) There exists a natural commutative diagram of profinite groups ρ brch n+1 Out FC (Π n+1 ) brch −−−→ Glu(Π n+1 ) ⏐ ⏐ ⏐ ⏐   ρ brch Out FC (Π n ) brch −− n −→ Glu(Π n ) [cf. Definition 4.6, (i); Definition 4.9; Lemma 4.10, (i); Defi- nition 4.11] — where the vertical arrows are injective. (ii) The closed subgroup Dehn(G) ⊆ (Aut(G) ⊆) Out(Π 1 ) [cf. [CbTpI], Definition 4.4] is contained in the image of the injection Out FC (Π n ) brch → Out FC (Π 1 ) brch [cf. the left-hand vertical ar- rows of the diagrams of (i), for varying n]. Thus, one may regard Dehn(G) as a closed subgroup of Out FC (Π n ) brch , i.e., Dehn(G) ⊆ Out FC (Π n ) brch . (iii) The homomorphism ρ brch : Out FC (Π n ) brch → Glu(Π n ) of (i) n and the inclusion Dehn(G) → Out FC (Π n ) brch of (ii) fit into an exact sequence of profinite groups ρ brch n 1 −→ Dehn(G) −→ Out FC (Π n ) brch −→ Glu(Π n ) −→ 1 . In particular, the commutative diagram of (i) is cartesian, and the horizontal arrows of this diagram are surjective. Proof. Assertion (i) follows immediately from Lemma 4.10, (i), together with the injectivity portion of [NodNon], Theorem B. Assertion (ii) follows immediately from Proposition 3.24, (ii); Theorem 4.2, (i). Finally, we verify assertion (iii). First, we claim that the following assertion holds: Claim 4.14.A: Ker(ρ brch n ) = Dehn(G) [cf. assertion (ii)]. 134 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Indeed, it follows immediately from Theorem 4.2, (iii) [cf. also Re- mark 4.9.1], together with assertion (i), that we have a natural com- mutative diagram ρ brch FC brch −− n −→ Glu(Π n ) 1 −−−→ Ker(ρ brch n ) −−−→ Out (Π n ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    ρ brch 1 −−−→ Dehn(G) −−−→ Out FC (Π 1 ) brch −− 1 −→ Glu(Π 1 ) −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are injective. Thus, Claim 4.14.A follows immediately. In particular, to complete the verification of assertion (iii), it suffices to verify the surjectivity of ρ brch n . The remainder of the proof of assertion (iii) is devoted to verifying this surjectivity. Next, we claim that the following assertion holds: Claim 4.14.B: If n = 2, then ρ brch is surjective. n We verify Claim 4.14.B by induction on #Node(G). If #Node(G) = 0, then Claim 4.14.B is immediate. If #Node(G) = 1, then Claim 4.14.B follows from Lemma 4.13. Now suppose that #Node(G) > 1, and that the induction hypothesis is in force. Let (α v ) v∈Vert(G) ∈ Glu(Π 2 ). Write ((α v ) 1 ) v∈Vert(G) ∈ Glu(Π 1 ) for the element of Glu(Π 1 ) determined by (α v ) v∈Vert(G) [i.e., the image of (α v ) v∈Vert(G) via the right-hand vertical arrow of the diagram of assertion (i) in the case where n = 1]. Let e ∈ Node(G). Write H for the unique sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of the underlying semi-graph of G whose set def of vertices is V(e). Then one verifies easily that S = Node(G| H ) \ {e} [cf. [CbTpI], Definition 2.2, (ii)] is not of separating type [cf. [CbTpI], Definition 2.5, (i)] as a subset of Node(G| H ). Thus, since (G| H ) S [cf. [CbTpI], Definition 2.5, (ii)] has precisely one node, and (α v ) v∈V(e) may be regarded as an element of Glu((Π H,S ) 2 ) — where we use the notation (Π H,S ) 2 to denote a configuration space subgroup of Π 2 associated to (H, S) [cf. Definition 4.3], to which the notation “Glu(−)” is applied in the evident sense — it follows from Lemma 4.13 that there exists an outomorphism β H,S of (Π H,S ) 2 ⊆ Π 2 that lifts (α v ) v∈V(e) ∈ Glu((Π H,S ) 2 ). Next, let us observe that it follows immediately from the various definitions involved that def γ = (β H,S , (α v ) v ∈V(e) ) ∈ Out((Π H,S ) 2 ) × Out((Π v ) 2 ) v ∈V(e) may be regarded as an element of the “Glu(Π 2 )” that occurs in the case where we take the stable log curve “X log ” to be a stable log curve over (Spec k) log obtained by deforming the node corresponding to e. Thus, since the number of nodes of such a stable log curve is = #Node(G) − 1 < #Node(G), by applying the induction hypothesis, we conclude that the above γ arises from an outomorphism α γ ∈ Out FC (Π 2 ) brch . COMBINATORIAL ANABELIAN TOPICS II 135 On the other hand, it follows immediately from the various definitions coincides with (α v ) v∈Vert(G) . This involved that the image of α γ via ρ brch 2 completes the proof of Claim 4.14.B. Finally, we verify the surjectivity of ρ brch [for arbitrary n] by in- n duction on n. If n ≤ 2, then the surjectivity of ρ brch follows from n Theorem 4.2, (iii) [cf. also Remark 4.9.1], Claim 4.14.B. Now sup- pose that n ≥ 3, and that the induction hypothesis is in force. Let (α v ) v∈Vert(G) ∈ Glu(Π n ). First, let us observe that it follows from the in- duction hypothesis that there exists an element α n−1 ∈ Out FC (Π n−1 ) brch such that ρ brch n−1 (α n−1 ) coincides with the element of Glu(Π n−1 ) deter-  n−1 be an mined by (α v ) v∈Vert(G) ∈ Glu(Π n ) [cf. assertion (i)]. Let α automorphism of Π n−1 that lifts α n−1 . Write α n−1/n−2 for the outomor- phism of Π n−1/n−2 determined by α  n−1 and α  n−2 for the automorphism  n−1 . of Π n−2 determined by α Next, let us observe that one verifies easily from the various defi- nitions involved that Π n/n−2 ⊆ Π n may be regarded as the “Π 2 ” as- sociated to some stable log curve “X log ” over (Spec k) log . Moreover, this stable log curve may be taken to be a geometric fiber of the sort discussed in Definition 3.1, (iii), in the case of the projection p log n−1/n−2 , relative to a point “x ∈ X n (k)” that maps to the interior of the same irreducible component of X log , relative to the n projections to X log . In particular, by fixing such a stable log curve, together with a suitable choice of lifting α  n−1 [cf. Theorem 4.7], it makes sense to speak of Glu(Π n/n−2 ). Moreover, it follows immediately from our choice of “x” that every configuration space subgroup that appears in the definition [cf. Definition 4.9, (ii)] of Glu(Π n/n−2 ) either • occurs as the intersection with Π n/n−2 of some configuration space subgroup that appears in the definition [cf. Definition 4.9, (iii)] of Glu(Π n ) or • projects isomorphically, via the projection Π n → Π 2 to the factors labeled n and n − 1, to a configuration space subgroup of Π 2 , i.e., a configuration space subgroup that appears in the definition [cf. Definition 4.9, (ii)] of Glu(Π 2 ). In particular, every tripod that appears in the definition [cf. Defini- tion 4.9, (ii)] of Glu(Π n/n−2 ) occurs as a tripod of a configuration space subgroup that appears either in the definition [cf. Definition 4.9, (iii)] of Glu(Π n ) or in the definition [cf. Definition 4.9, (ii)] of Glu(Π 2 ). Moreover, it follows from Theorem 4.7; Lemma 3.2, (iv); Lemma 4.8, (i), that the various α v ’s preserve the conjugacy classes of these config- uration space subgroups and tripods — as well as each conjugacy class of cuspidal inertia subgroups of each of these tripods! — that appear in the definition [cf. Definition 4.9, (ii)] of Glu(Π n/n−2 ). Thus, we con- clude from Theorem 3.18, (ii), together with Definition 4.9, (iii), in the case of Glu(Π n ), and Definition 4.9, (ii), in the case of Glu(Π 2 ), that 136 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (α v ) v∈Vert(G) determines an element ∈ Glu(Π n/n−2 ), hence, by Claim 4.14.B, an element α n/n−2 ∈ Out FC (Π n/n−2 ) that lifts the element α n−1/n−2 ∈ Out(Π n−1/n−2 ). Now we claim that the following assertion holds: Claim 4.14.C: This outomorphism α n/n−2 of Π n/n−2 is compatible with the automorphism α  n−2 of Π n−2 rel- ative to the homomorphism Π n−2 → Out(Π n/n−2 ) in- duced by the natural exact sequence of profinite groups p Π n/n−2 1 −→ Π n−2/n −→ Π n −→ Π n−2 −→ 1 . Indeed, this follows immediately from the corresponding fact for α n−1/n−2 [which follows from the existence of α  n−1 ], together with the injectivity FC of the natural homomorphism Out (Π n/n−2 ) → Out FC (Π n−1/n−2 ) [cf. [NodNon], Theorem B]. This completes the proof of Claim 4.14.C. ∼ Thus, by applying Claim 4.14.C and the natural isomorphism Π n → out Π n/n−2  Π n−2 [cf. the discussion entitled “Topological groups” in [CbTpI], §0], we obtain an outomorphism α n of Π n that lifts the outo- morphism α n−1 of Π n−1 . Thus, it follows immediately from Lemma 4.10, (i), that ρ brch n (α n ) = (α v ) v∈Vert(G) . This completes the proof of the sur-  jectivity of ρ brch n , hence also of assertion (iii). Remark 4.14.1. In the notation of Theorem 4.14, observe that the data of collections of smooth log curves that [by gluing at prescribed cusps] give rise to a stable log curve whose associated semi-graph of anabelioids [of pro-Σ PSC-type] is isomorphic to G form a smooth, connected moduli stack. In particular, by considering a suitable path in the étale fundamental groupoid of this moduli stack, one verifies immediately that one may reduce the verification of an “isomorphism version” — i.e., concerning PFC-admissible [cf. [CbTpI], Definition 1.4, (iii)] outer isomorphisms between the pro-Σ fundamental groups of the configuration spaces associated to two a priori distinct stable log curves “X log ” and “Y log ” — of Theorem 4.14 to the “automor- phism version” given in Theorem 4.14 [cf. [CmbCsp], Remark 4.1.4]. A similar statement may be made concerning Theorem 4.7. We leave the routine details to the interested reader. In the present monograph, we restricted our attention to the “automorphism versions” of these results in order to simplify the [already somewhat complicated!] nota- tion. Remark 4.14.2. One may regard [CmbCsp], Corollary 3.3, as a special discussed in Theorem 4.14, i.e., the case case of the surjectivity of ρ brch n COMBINATORIAL ANABELIAN TOPICS II 137 in which X log is obtained by gluing a tripod to a smooth log curve along a cusp of the smooth log curve. Corollary 4.15 (Surjectivity result). In the notation of Theorem 3.16, suppose that n ≥ 3. If r = 0, then we suppose further that n ≥ 4. Then the tripod homomorphism T Π tpd : Out F (Π n ) −→ Out C (Π tpd ) Δ+ [cf. Definition 3.19] is surjective. Proof. Let α ∈ Out C (Π tpd ) Δ+ . First, let us observe that — by consid- ering a suitable stable log curve of type (g, r) over (Spec k) log and ap- plying a suitable specialization isomorphism [cf. Proposition 3.24, (i); the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] — to verify Corollary 4.15, we may assume without loss of generality that G is totally degenerate [cf. [CbTpI], Definition 2.3, (iv)], i.e., that every vertex of G is a tripod of X n log [cf. Defini- tion 3.1, (v)]. Then since α ∈ Out C (Π tpd ) Δ+ , it follows immediately from [CmbCsp], Corollary 4.2, (i), (ii) [cf. also [CmbCsp], Definition 1.11, (i)], that there exists an element α n ∈ Out FC (Π tpd n ) — where we for the “Π ” that occurs in the case where we take “X log ” write Π tpd n n to be a tripod — such that α arises as the image of α n via the nat- FC tpd ural injection Out FC (Π tpd ) of [NodNon], Theorem B. n ) → Out (Π Thus, it follows immediately from Theorem 4.14, (iii), that there ex- — the ists an element β ∈ Out FC (Π n ) brch that lifts — relative to ρ brch n element of Glu(Π n ) [cf. Theorems 3.16, (v); 3.18, (ii)] determined by α n ∈ Out FC (Π tpd n ). [Here, recall that we have assumed that G is totally degenerate.] Finally, it follows from Theorems 3.16, (v); 3.18, (ii), that T Π tpd (β) = α, i.e., that α is contained in the image of T Π tpd . This completes the proof of Corollary 4.15.  Corollary 4.16 (Absolute anabelian cuspidalization for stable log curves over finite fields). Let p, l X , l Y be prime numbers such that p ∈ {l X , l Y }; (g X , r X ), (g Y , r Y ) pairs of nonnegative integers such that 2g X −2+r X , 2g Y −2+r Y > 0; k X , k Y finite fields of characteristic p; k X , k Y algebraic closures of k X , k Y ; (Spec k X ) log , (Spec k Y ) log the log schemes obtained by equipping Spec k X , Spec k Y with the log structures determined by the fs charts N → k X , N → k Y that map 1 → 0; X log , Y log stable log curves of type (g X , r X ), (g Y , r Y ) over (Spec k X ) log , (Spec k Y ) log ; def def def def log G log k X = π 1 ((Spec k X ) )  G k X = Gal(k X /k X ) , log G log k Y = π 1 ((Spec k Y ) )  G k Y = Gal(k Y /k Y ) 138 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI the natural surjections [well-defined up to composition with an inner log automorphism]; s X : G k X → G log k X , s Y : G k Y → G k Y sections of the log above natural surjections G log k X  G k X , G k Y  G k Y . For each positive integer n, write X n log , Y n log for the n-th log configuration spaces [cf. the discussion entitled “Curves” in “Notations and Conventions”] of X log , Y log ; X Π n , Y Π n for the maximal pro-l X , pro-l Y quotients of the log log kernels of the natural surjections π 1 (X n log )  G log k X , π 1 (Y n )  G k Y . Then the sections s X , s Y determine outer actions of G k X , G k Y on X Π n , Y Π n . Thus, we obtain profinite groups out out Π n  s X G k X , Y Π n  s Y G k Y X [cf. [MzTa], Proposition 2.2, (ii); the discussion entitled “Topological groups” in [CbTpI], §0]. Let out out ∼ α 1 : X Π 1  s X G k X −→ Y Π 1  s Y G k Y be an isomorphism of profinite groups. Then l X = l Y ; there exists a unique collection of isomorphisms of profinite groups   out out ∼ α n : X Π n  s X G k X −→ Y Π n  s Y G k Y n≥1 out — well-defined up to composition with an inner automorphism of Y Π n  s Y G k Y by an element of the intersection Y Ξ n ⊆ Y Π n of the fiber subgroups of Y Π n of co-length 1 [cf. [CmbCsp], Definition 1.1, (iii)] — such that each diagram out X α n+1 out Π n+1  s X G k X −−−→ Y Π n+1  s Y G k Y ⏐ ⏐ ⏐ ⏐   X out Π n  s X G k X α n −−− → Y out Π n  s Y G k Y — where the vertical arrows are the surjections induced by the pro- log log jections X n+1 → X n log , Y n+1 → Y n log obtained by forgetting the factors labeled j, for some j ∈ {1, · · · , n+1} — commutes, up to composition with a Y Ξ n -inner automorphism. Proof. First, let us observe that it follows from Corollary 4.18, (ii), def below that l X = l Y . Write l = l X = l Y . Moreover, it follows from Corollary 4.18, (viii), below [i.e., in the case where condition (viii-1) out is satisfied] that α 1 maps X Π 1 ⊆ X Π 1  s X G k X bijectively onto Y Π 1 ⊆ Y out Π 1  s Y G k Y . In particular, α 1 induces isomorphisms of profinite groups ∼ ∼ α 1 Π : X Π 1 −→ Y Π 1 , α 0 : G k X −→ G k Y . For  ∈ {X, Y }, write G  for the semi-graph of anabelioids of pro-l PSC-type determined by  log ; Π G  for the [pro-l] fundamental group COMBINATORIAL ANABELIAN TOPICS II 139 (l) of G  ; G k  ⊆ G k  for the maximal pro-l closed subgroup of G k  ; ( =l) G k  for the maximal pro-prime-to-l closed subgroup of G k  . Thus, we have a natural π 1 ( log )-orbit, i.e., relative to composition with au- tomorphisms induced by conjugation by elements of π 1 ( log ), of iso- ∼ ∼ morphisms  Π 1 → Π G  ; fix an isomorphism  Π 1 → Π G  that belongs to the collection of isomorphisms that constitutes this π 1 ( log )-orbit of isomorphisms. Moreover, since G k  is isomorphic to Z as an abstract profinite group, we have a natural decomposition (l) ( =l) ∼ G k  × G k  −→ G k  . Thus, the isomorphism α 0 naturally decomposes into a pair of isomor- phisms (l) (l) ∼ (l) ( =l) α 0 : G k X −→ G k Y , α 0 ( =l) ∼ ( =l) : G k X −→ G k Y . Next, let us observe that since  Π 1 is topologically finitely generated [cf. [MzTa], Proposition 2.2, (ii)] and pro-l, one verifies easily that [by replacing G k  by a suitable open subgroup and applying the injectivity portion of [NodNon], Theorem B, together with [CmbGC], Corollary 2.7, (i)] we may assume without loss of generality that the outer action of G k  on  Π 1 — hence [cf. the injectivity portion of [NodNon], Theo- rem B] also on  Π n for each positive integer n — factors through the ∼ (l) ( =l) (l) quotient G k  ← G k  × G k   G k  . Next, let us recall the following well-known Facts: (1) Some positive tensor power of the l-adic cyclotomic character of G k  factors through the outer action of G k  on  Π 1 [cf. Corollary 4.18, (vii), below]. (l) (2) The restriction to G k  ⊆ G k  of any positive tensor power of the l-adic cyclotomic character of G k  is injective. Thus, it follows from Facts (1), (2), that (3) the resulting outer action of G k  on  Π 1 — hence also on  Π n for each positive integer n — is injective. (l) In particular, it follows immediately from the slimness of  Π n [cf. [MzTa], Proposition 2.2, (ii)] that the composite Z  out out Π n  s  G k  (  Π n ) →  Π n  s  G k   G k  determines an isomorphism Z  ∼ out Π n  s  G k  (  Π n ) −→ G =l k  . 140 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Thus, if we identify Z  out Π n  s  G k  (  Π n ) with G =l k  by means of this iso- morphism, then we obtain a natural isomorphism   out out (l) ( =l) ∼  Π n  s  G k  × G k  −→  Π n  s  G k  . Next, let us observe that the following assertion holds: Claim 4.16.A: There exists a positive power q of p such that log p (q) is divisible by log p (#k X ), log p (#k Y ), and, moreover, (l) (l) (l) α 0 ((Fr q ) k X ) = (Fr q ) k Y — where we write (Fr q ) k X ∈ G k X , (Fr q ) k Y ∈ G k Y for (l) the q-power Frobenius elements of G k X , G k Y ; (Fr q ) k X ∈ (l) (l) (l) G k X , (Fr q ) k Y ∈ G k Y for the respective images of (Fr q ) k X ∈ (l) (l) G k X , (Fr q ) k Y ∈ G k Y in G k X , G k Y . Indeed, this follows immediately from Corollary 4.18, (vii), below, to- gether with Fact (2). Write H k X ⊆ G k X , H k Y ⊆ G k Y for the open subgroups of G k X , G k Y topologically generated by (Fr q ) k X ∈ G k X , (Fr q ) k Y ∈ G k Y [cf. Claim 4.16.A]; U k Y ⊆ G k Y for the open subgroup of G k Y topologically gener- (l) (l) ated by α 0 ((Fr q ) k X ) ∈ G k Y ; H k X ⊆ G k X for the image of H k X ⊆ G k X in (l) (l) (l) (l) (l) G k X ; H k Y , U k Y ⊆ G k Y for the images of H k Y , U k Y ⊆ G k Y in G k Y . Then (l) (l) it follows from Claim 4.16.A that we have an equality H k Y = U k Y , and, ∼ moreover, that the isomorphism H k X → U k Y induced by α 0 induces an (l) ∼ (l) (l) isomorphism H k X → U k Y = H k Y . In particular, one verifies easily that ∼ there exists an isomorphism of profinite groups α 0 H : H k X → H k Y that (a) maps (Fr q ) k X ∈ G k X to (Fr q ) k Y ∈ G k Y , which thus implies that ∼ (b) the isomorphism H k X → H k Y induced by α 0 H coincides with (l) ∼ (l) (l) the above isomorphism H k X → U k Y = H k Y induced by α 0 . (l) (l) Moreover, it follows immediately from (b), together with the existence of the natural isomorphisms   out out (l) ( =l) ∼ X Π n  s X G k X × G k X −→ X Π n  s X G k X ,   out out (l) ( =l) ∼ Y Π n  s Y G k Y × G k Y −→ Y Π n  s Y G k Y [cf. the discussion preceding Claim 4.16.A], that there exists an iso- morphism out ∼ out α 1 H : X Π 1  s X H k X −→ Y Π 1  s Y H k Y such that COMBINATORIAL ANABELIAN TOPICS II 141 (c) the isomorphism “α 0 ” of H k X with H k Y that occurs in the case where we take the “α 1 ” to be α 1 H coincides with α 0 H [i.e., roughly speaking, α 1 H lies over α 0 H ], and, moreover, (d) the isomorphism “α 1 Π ” of X Π 1 with Y Π 1 that occurs in the case where we take the “α 1 ” to be α 1 H coincides with [the original] α 1 Π [i.e., roughly speaking, α 1 H restricts to α 1 Π on X Π 1 ]. In particular, we conclude, again by the existence of the natural iso- morphisms   out out (l) ( =l) ∼ X Π n  s X G k X × G k X −→ X Π n  s X G k X ,   out out (l) ( =l) ∼ Y Π n  s Y G k Y × G k Y −→ Y Π n  s Y G k Y , together with the injectivity portion of [NodNon], Theorem B, and [CmbGC], Corollary 2.7, (i), that, to verify Corollary 4.16 — by re- placing G k X , G k Y , α 1 by H k X , H k Y , α 1 H — we may assume without loss of generality that #k X = #k Y , and that α 0 maps the #k X -power Frobenius element of G k X to the #k Y -power Frobenius element of G k Y . We may also assume without loss of generality — by replacing G k  , where  ∈ {X, Y }, by a suitable open subgroup of G k  if necessary — that the following condition holds: (e) for  ∈ {X, Y }, G k  acts trivially on the underlying semi- graph of G  . Next, let us observe that the uniqueness portion of Corollary 4.16 follows immediately from the injectivity portion of [NodNon], Theorem B, and [CmbGC], Corollary 2.7, (i). Thus, it remains to verify the existence of a collection of α n ’s as in the statement of Corollary 4.16. To this end, for each positive integer i,  ∈ {X, Y }, and v ∈ Vert(G  ), write (  Π v ) i ⊆  Π i for the configuration space subgroup of  Π i as- sociated to v ∈ Vert(G  ) [well-defined up to  Π i -conjugation — cf. Definition 4.3]. Next, let us observe that ∼ (f) the isomorphism Π G X → Π G Y determined by α 1 Π and the fixed ∼ ∼ isomorphisms X Π 1 → Π G X , Y Π 1 → Π G Y is graphic [cf. Corol- lary 4.18, (iii), (iv), below]. ∼ Write α Vert : Vert(G X ) → Vert(G Y ) for the bijection determined by ∼ the graphic isomorphism Π G X → Π G Y of (f). Thus, for each v ∈ ∼ Vert(G X ), the isomorphism Π G X → Π G Y of (f) determines an outer iso- ∼ morphism β v : ( X Π v ) 1 → ( Y Π α Vert (v) ) 1 [cf. [CmbGC], Proposition 1.2, (ii); [CbTpI], Lemma 2.12, (i), (ii), (iii)], which is compatible with the respective natural outer actions of G k X , G k Y [cf. (e)]. In par- ticular, by applying [Wkb], Theorem C, to this outer isomorphism 142 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ β v : ( X Π v ) 1 → ( Y Π α Vert (v) ) 1 , we obtain [cf. [CmbGC], Corollary 2.7, (i)] a PFC-admissible [cf. [CbTpI], Definition 1.4, (iii)] outomorphism ∼ β v,n : ( X Π v ) n → ( Y Π α Vert (v) ) n , which is compatible with the respective natural outer actions of G k X , G k Y [cf. (e)]. Moreover, since the β v ’s ∼ arise from a single isomorphism Π G X → Π G Y , one verifies immediately from [CbTpI], Corollary 3.9, (ii), (v), and the injectivity discussed in [Hsh], Remark 6, (iv) [i.e., applied to the difference between the vari- ous outer isomorphisms, determined by β v,n , between tripods of ( X Π v ) n and tripods of ( Y Π α Vert (v) ) n ], that the collection (β v,n ) v∈Vert(G X ) is con- tained in the set which corresponds — in the “isomorphism version” of Theorem 4.14 discussed in Remark 4.14.1 — to the set “Glu(Π n )” in the statement of Theorem 4.14. In particular, it follows from the “isomorphism version” of Theorem 4.14, (i), (iii), discussed in Re- mark 4.14.1 that the outer isomorphism determined by the isomor- ∼ phism α 1 Π : X Π 1 → Y Π 1 and the collection (β v,n ) v∈Vert(G) uniquely deter- ∼ mine a PFC-admissible outer isomorphism β n : X Π n → Y Π n which — by the injectivity portion of [NodNon], Theorem B — is compatible with the respective outer actions of G k X , G k Y . Finally, one verifies immedi- ately that one may construct a collection of α n ’s as in the statement of Corollary 4.16 from the collection of the β n ’s. This completes the  proof of the existence of α n ’s, hence also of Corollary 4.16. Remark 4.16.1. Corollary 4.16 may be regarded as a generalization of [AbsCsp], Theorem 3.1; [Hsh], Theorem 0.1; [Wkb], Theorem C. Corollary 4.17 (Commensurator of the image of the absolute Galois group of a finite field in the totally degenerate case). Let n be a positive integer; p, l two distinct prime numbers; (g, r) a pair of nonnegative integers  = (0, 3) such that 2g −2+r > 0; k a finite field of characteristic p; k an algebraic closure of k; (Spec k) log the log scheme obtained by equipping Spec k with the log structure determined by the fs chart N → k that maps 1 → 0; X log a stable log curve of type (g, r) over (Spec k) log . Write G for the semi-graph of anabelioids of pro-l PSC-type associated to the stable log curve X log ; G for the underlying semi-graph of G; Π G for the [pro-l] fundamental group of G; def def log G log k = π 1 ((Spec k) )  G k = Gal(k/k) for the natural surjection [well-defined up to composition with an inner automorphism]. For each positive integer i, write X i log for the i-th log configuration space [cf. the discussion entitled “Curves” in “No- tations and Conventions”] of X log ; Π i for the maximal pro-l quotient of the kernel of the natural surjection π 1 (X i log )  G log k . Thus, we COMBINATORIAL ANABELIAN TOPICS II 143 have a natural π 1 (X log )-orbit, i.e., relative to composition with auto- morphisms induced by conjugation by elements of π 1 (X log ), of isomor- ∼ phisms Π 1 → Π G and a natural outer action FC ρ X log : G log k −→ Out (Π i ) i [cf. the notation of [CmbCsp], Definition 1.1, (ii)]. Fix an outer ∼ isomorphism Π 1 → Π G whose constituent isomorphisms belong to the above π 1 (X log )-orbit of isomorphisms. Let H ⊆ G log k be a closed sub- log group of G k whose image in G k is open. Write I H ⊆ H for the kernel of the composite H → G log k  G k . We shall say that H is of l-Dehn type if the maximal pro-l quotient of I H is nontrivial. Suppose that the stable log curve X log is totally degenerate [i.e., that the com- plement in X of the nodes and cusps is a disjoint union of tripods]. Then the following hold: (i) The image ρ X log (I H ) ⊆ Out(Π 1 ) is contained in Dehn(G) ⊆ 1 ∼ Out(Π G ) ← Out(Π 1 ) [cf. the notation of [CbTpI], Definition 4.4]. Moreover, the image ρ X log (I H ) is nontrivial if and only 1 if H is of l-Dehn type. Write C(ρ) def I H = (ρ X log (I H ) ⊗ Z l Q l ) ∩ Dehn(G) ⊆ Dehn(G) 1 [considered in Dehn(G) ⊗ Z l Q l — cf. [CbTpI], Theorem 4.8, (iv)]. (ii) For any positive integer m ≤ n, the natural injection Out FC (Π n ) → Out FC (Π m ) of [NodNon], Theorem B, induces isomor- phisms ∼ Z Out FC (Π n ) (ρ X n log (H)) −→ Z Out FC (Π m ) (ρ X m log (H)) , ∼ loc loc Z Out FC (Π ) (ρ X log (H)) −→ Z Out FC (Π ) (ρ X log (H)) n m n m [cf. the discussion entitled “Topological groups” in “Notations and Conventions”], ∼ N Out FC (Π n ) (ρ X n log (H)) −→ N Out FC (Π m ) (ρ X m log (H)) , ∼ C Out FC (Π n ) (ρ X n log (H)) −→ C Out FC (Π m ) (ρ X m log (H)) . ∼ (iii) Relative to the natural inclusion Aut(G) (⊆ Out(Π G ) ← Out(Π 1 )), the following equality holds: C Out FC (Π 1 ) (ρ X log (H)) = C Aut(G) (ρ X log (H)) . 1 1 In particular, we have natural homomorphisms of profinite groups ∼ C Out FC (Π n ) (ρ X n log (H)) → C Out FC (Π 1 ) (ρ X log (H)) → Aut(G) , 1 χ G ∼ C Out FC (Π n ) (ρ X n log (H)) → C Out FC (Π 1 ) (ρ X log (H)) → Z ∗ l 1 144 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. the notation of [CbTpI], Definition 3.8, (ii)] — where the first arrow in each line is the isomorphism of (ii). By abuse of notation [i.e., since ρ X n log (H) is not necessarily contained in Aut |grph| (G) — cf. the notation of [CbTpI], Definition 2.6, (i); Remark 4.1.2 of the present monograph], write Z Aut |grph| (G) (ρ X n log (H)) ⊆ Z Out FC (Π n ) (ρ X n log (H)) , loc loc Z Aut |grph| (G) (ρ X log (H)) ⊆ Z Out FC (Π ) (ρ X log (H)) , n n n N Aut |grph| (G) (ρ X n log (H)) ⊆ N Out FC (Π n ) (ρ X n log (H)) , C Aut |grph| (G) (ρ X n log (H)) ⊆ C Out FC (Π n ) (ρ X n log (H)) for the kernels of the restrictions of the composite homomor- phism of the first line of the second display [of the present (iii)] to loc Z Out FC (Π n ) (ρ X n log (H)), Z Out FC (Π ) (ρ X log (H)) , n n N Out FC (Π n ) (ρ X n log (H)), C Out FC (Π n ) (ρ X n log (H)) , respectively. (iv) Suppose that H is not of l-Dehn type. Then we have equal- ities loc Z Aut |grph| (G) (ρ X n log (H)) = Z Aut |grph| (G) (ρ X log (H)) n = N Aut |grph| (G) (ρ X n log (H)) = C Aut |grph| (G) (ρ X n log (H)) [cf. the notation of (iii)]. Moreover, each of the four groups appearing in these equalities is, in fact, independent of n [cf. (ii)]. (v) Suppose that H is of l-Dehn type. Then the composite ho- momorphism of the first line of the second display of (iii) de- termines an injection of profinite groups loc Z Out FC (Π ) (ρ X log (H)) → Aut(G) . n n (vi) Write k |grph| (⊆ k) for the [finite] subfield of k consisting of the invariants of k with respect to [the natural action on k of ] the kernel of the natural action of H on G. Then the composite homomorphism of the second line of the second display of (iii) determines natural exact sequences of profinite groups N (ρ) −→ N Aut |grph| (G) (ρ X n log (H)) −→ Z ∗ l , C(ρ) −→ C Aut |grph| (G) (ρ X n log (H)) −→ Z ∗ l 1 −→ I H 1 −→ I H [cf. the notation of (i), (iii)] — where ρ X n log (I H ), hence also N (ρ) def (ρ X n log (I H ) ⊆) I H = N Aut |grph| (G) (ρ X n log (H)) ∩ Dehn(G) COMBINATORIAL ANABELIAN TOPICS II 145 C(ρ) [cf. (i), (ii), (iii)], is an open subgroup of I H ; the image of the third arrow in each line contains #k |grph| ∈ Z ∗ l and does not depend on the choice of n. In particular, these images are open; if, moreover, #k |grph| ∈ Z ∗ l topologically generates Z ∗ l , then the third arrows in each line are surjective. (vii) The closed subgroup ρ X n log (H) ⊆ C Out FC (Π n ) (ρ X n log (H)), hence also N Out FC (Π n ) (ρ X n log (H)) (⊆ C Out FC (Π n ) (ρ X n log (H))), is open in C Out FC (Π n ) (ρ X n log (H)). (viii) Consider the following conditions [cf. Remark 4.17.1 below]: (1) Write Aut (Spec k) log (X log ) for the group of automorphisms of X log over (Spec k) log . Then the natural homomorphism Aut (Spec k) log (X log ) −→ Aut(G) is surjective. (2) #k |grph| ∈ Z ∗ l topologically generates Z ∗ l . If condition (1) is satisfied, and H is of l-Dehn type, then we have an equality loc Z Out FC (Π n ) (ρ X n log (H)) = Z Out FC (Π ) (ρ X log (H)) , n n and, moreover, the composite homomorphism of the first line of the second display of (iii) determines an isomorphism ∼ loc Z Out FC (Π ) (ρ X log (H)) −→ Aut(G) . n n If conditions (1) and (2) are satisfied, then the composite ho- momorphisms of the two lines of the second display of (iii) determine natural exact sequences of profinite groups N (ρ) −→ N Out FC (Π n ) (ρ X n log (H)) −→ Aut(G) × Z ∗ l −→ 1 , C(ρ) −→ C Out FC (Π n ) (ρ X n log (H)) −→ Aut(G) × Z ∗ l −→ 1 . 1 −→ I H 1 −→ I H Proof. Assertion (i) follows immediately from the various definitions involved, together with [CbTpI], Lemma 5.4, (ii); [CbTpI], Proposi- tion 5.6, (ii). Assertion (ii) follows immediately from Corollary 4.16, together with the slimness of Π i for each positive integer i [cf. [MzTa], Proposition 2.2, (ii)] and the openness of the image of H in G k . As- sertion (iii) follows immediately from [CmbGC], Corollary 2.7, (ii) [cf. also the proof of [CmbGC], Proposition 2.4, (v)], together with the openness of the image of H in G k . For  ∈ {Z, Z loc , N, C} and v ∈ Vert(G), write ∼ def  =  Out FC (Π 1 ) (ρ X log (H)) ⊆ Out(Π 1 ) → Out(Π G ) ; 1  |grph| =  ∩ Aut |grph| (G) ⊆ Out(Π G ) def 146 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. the notation of [CbTpI], Definition 2.6, (i); Remark 4.1.2 of the present monograph]; pr v : Aut |grph| (G) −→ Aut |grph| (G| v ) for the homomorphism determined by restriction to G| v [cf. [CbTpI], Definition 2.14, (ii); [CbTpI], Remark 2.5.1, (ii)];  v ⊆ Aut |grph| (G| v ) for the image of  |grph| ⊆ Aut |grph| (G) via pr v . Then we claim that the following assertion holds: Claim 4.17.A: Let v ∈ Vert(G). Then C v ∩ Ker(χ G| v ) = {1} [cf. the notation of [CbTpI], Definition 3.8, (ii)]. Indeed, let us first observe that since  Π 1 is topologically finitely gen- erated [cf. [MzTa], Proposition 2.2, (ii)] and pro-l, one verifies easily that the image of the outer action ρ X log admits a pro-l open subgroup. 1 Thus, since the image of H in G k is open, it follows immediately from Corollary 4.18, (vii), below that C v ⊆ Aut |grph| (G| v ) is contained in the local centralizer [cf. the discussion entitled “Topological groups” in “No- tations and Conventions”] of the natural image of G k in Aut |grph| (G| v ) [cf. the fact that G| v is of type (0, 3)]. Thus, Claim 4.17.A follows im- mediately from the injectivity discussed in [Hsh], Remark 6, (iv). This completes the proof of Claim 4.17.A. Next, we claim that the following assertion holds: Claim 4.17.B: Let v ∈ Vert(G). Then C |grph| ∩ Ker(pr v ) = C |grph| ∩ Ker(χ G ) = C |grph| ∩ Dehn(G) ; loc ∩ Ker(pr v ) = {1} . Z |grph| ∩ Ker(pr v ) = Z |grph| In particular, we obtain natural isomorphisms ∼ ∼ loc Z |grph| −→ Z v , Z |grph| −→ Z v loc and a natural exact sequence of profinite groups χ G 1 −→ C |grph| ∩ Dehn(G) −→ C |grph| −→ Z ∗ l . Indeed, let us first observe that the equalities of the first line of the first display of Claim 4.17.B follow immediately from Claim 4.17.A, together with [CbTpI], Corollary 3.9, (iv). Moreover, since the image of H in G k is open, the equalities of the second line of the first display of Claim 4.17.B follow immediately from [CbTpI], Theorem 4.8, (iv), (v), together with the equalities of the first line of the first display of Claim 4.17.B. This completes the proof of Claim 4.17.B. Next, we verify assertion (iv). Let us first observe that it follows from assertion (ii) that it suffices to verify assertion (iv) in the case where n = 1. Next, let us observe that it follows from Lemma 3.9, COMBINATORIAL ANABELIAN TOPICS II 147 (ii), that C |grph| ⊆ N Out FC (Π 1 ) (Z loc ), which thus implies that we have a loc natural action [by conjugation] of C |grph| on Z loc , hence also on Z |grph| , as well as a natural [trivial!] action of C |grph| on Aut(G). Moreover, by considering the inclusion ∼ loc (C |grph| ⊇) Z |grph| → Z v loc → Z ∗ l induced by χ G| v [cf. Claims 4.17.A, 4.17.B], we conclude that the ho- momorphisms of the two lines of the second display of assertion (iii) determine a natural [C |grph| -equivariant!] injection Z loc → Aut(G) × Z ∗ l . Thus, since Z ∗ l is abelian, it follows that C |grph| acts trivially on Z loc , i.e., that C |grph| ⊆ Z Out FC (Π 1 ) (Z loc ). On the other hand, since H is not of l-Dehn type, one verifies easily from assertion (i) that ρ X log (H) is 1 abelian, hence that ρ X log (H) ⊆ Z ⊆ Z loc . Thus, we conclude that 1 C |grph| ⊆ Z Out FC (Π 1 ) (Z loc ) ∩ Aut |grph| (G) ⊆ Z Out FC (Π 1 ) (ρ X log (H)) ∩ Aut |grph| (G) 1 = Z ∩ Aut |grph| (G) = Z |grph| . This completes the proof of assertion (iv). Next, we verify assertion (v). First, let us observe that it follows from assertion (ii) that, to verify assertion (v), it suffices to verify that loc loc Z |grph| = {1}, hence, by Claim 4.17.B, that χ G (Z |grph| ) = {1}. On the other hand, since H is of l-Dehn type, by considering the conjugation loc action of Z |grph| on ρ X log (I H ) [which is nontrivial by assertion (i)], we 1 loc ) = {1}, conclude from [CbTpI], Theorem 4.8, (iv), (v), that χ G (Z |grph| as desired. This completes the proof of assertion (v). Next, we verify assertion (vi). First, we observe that it follows from N (ρ) assertions (ii), (iii) that the definition of I H is indeed independent of n [as the notation suggests!]. Next, we claim that the following assertion holds: Claim 4.17.C: N (ρ) ρ X log (I H ) ⊆ N |grph| ∩ Dehn(G) = I H 1 C(ρ) ⊆ C |grph| ∩ Dehn(G) = I H . Indeed, the final equality follows immediately from an elementary com- putation [in which we apply [CbTpI], Theorem 4.8, (iv), (v)], together with assertion (i); the remainder of Claim 4.17.C follows immediately from the various definitions involved, together with assertion (i). This completes the proof of Claim 4.17.C. Now it follows immediately from Claims 4.17.B, 4.17.C, together with assertion (ii), that the composite homomorphism of the second line of the second display of (iii) deter- mines the two displayed exact sequences of assertion (vi), and that N (ρ) C(ρ) ρ X log (I H ), hence also I H , is an open subgroup of I H . Moreover, 1 148 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI since [it is immediate that] the image, via ρ X n log , of the kernel of the natural action of H on G is contained in N |grph| , the image of the third arrow in each line of the displayed sequences of assertion (vi) contains #k |grph| ∈ Z ∗ l . Finally, it follows from assertion (ii) that the image of the third arrow in each line of the displayed sequences of assertion (vi) does not depend on the choice of n. This completes the proof of assertion (vi). Assertion (vii) follows immediately from assertions (iii) and (vi), together with the finiteness of Aut(G). Assertion (viii) follows im- mediately from assertions (v) and (vi). This completes the proof of Corollary 4.17.  Remark 4.17.1. (i) One verifies easily that condition (1) of Corollary 4.17, (viii), holds if, for instance, k = k |grph| , and, moreover, the lengths [cf. [CbTpI], Definition 5.3, (ii)] of the various nodes of X log [whose base-change from k to k may be thought of as the special fiber stable log curve of [CbTpI], Definition 5.3] coincide. (ii) In a similar vein, one verifies easily that condition (2) of Corol- lary 4.17, (viii), holds if, for instance, k |grph| = F p , and, more- 2 over, p remains prime in the cyclotomic extension Q(e 2πi/l ), √ where i = −1, and we assume that l is odd. Remark 4.17.2. The computation, in the case where n = 1, of the cen- tralizer (respectively, normalizer and commensurator) in Corollary 4.17, (viii), may be thought of as a sort of relative geometrically pro-l (respectively, [semi-] absolute geometrically pro-l) version of the Grothendieck Conjecture for totally degenerate stable log curves over finite fields. In fact, the proofs of these computations of Corol- lary 4.17, (viii), in the case where n = 1, only involve the theory of [CmbGC] and [CbTpI]. On the other hand, these computations of Corollary 4.17, (viii), can only be performed under certain relatively restrictive conditions [cf. Remark 4.17.1]. It is precisely for this reason that Corollary 4.17, (ii), which may be thought of as an application of the theory of the present monograph, is of interest in the context of these computations of Corollary 4.17, (viii). Corollary 4.18 (Compatibility with geometric subgroups). Let p, l X , l Y be prime numbers such that p ∈ {l X , l Y }; (g X , r X ), (g Y , r Y ) pairs of nonnegative integers such that 2g X − 2 + r X , 2g Y − 2 + r Y > 0; k X , k Y finite fields of characteristic p; k X , k Y algebraic closures of k X , k Y ; (Spec k X ) log , (Spec k Y ) log the log schemes obtained by equipping COMBINATORIAL ANABELIAN TOPICS II 149 Spec k X , Spec k Y with the log structures determined by the fs charts N → k X , N → k Y that map 1 → 0; X log , Y log stable log curves of type (g X , r X ), (g Y , r Y ) over (Spec k X ) log , (Spec k Y ) log ; def def def def log G log k X = π 1 ((Spec k X ) )  G k X = Gal(k X /k X ) , log G log k Y = π 1 ((Spec k Y ) )  G k Y = Gal(k Y /k Y ) the natural surjections [well-defined up to composition with an inner log log log Y automorphism]; X H ⊆ G log k X , H ⊆ G k Y closed subgroups of G k X , G k Y ; X I ⊆ X H, Y I ⊆ Y H the kernels of the composites X H → G log k X  G k X , log Y X Y H → G k Y  G k Y ; Π, Π the maximal pro-l X , pro-l Y quotients of log )  G log the kernels of the natural surjections π 1 (X log )  G log k X , π 1 (Y k Y ; G X , G Y the semi-graphs of anabelioids of pro-l PSC-type determined by X log , Y log ; Π G X , Π G Y the [pro-l] fundamental groups of G X , G Y [so we have natural π 1 (X log )-, π 1 (Y log )-orbits — i.e., relative to composition with automorphisms induced by conjugation by elements of π 1 (X log ), ∼ ∼ π 1 (Y log ) — of isomorphisms X Π → Π G X , Y Π → Π G Y ]. Then the natural log X Y X outer actions of G log k X , G k Y on Π, Π determine outer actions of I ⊆ X H, Y I ⊆ Y H on X Π, Y Π. Thus, we obtain profinite groups X out out Π  X I ⊆ X Π  X H, Y out out Π  Y I ⊆ Y Π  Y H [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. Sup- pose that, for each  ∈ {X, Y }, one of the following two conditions is satisfied: (a) The equality  H = G log k  holds. (b) The composite  H → G log k   G k  is an isomorphism. We shall refer to a closed subgroup of X Π, Y Π obtained by forming the image — by the inverse of an element of the π 1 (X log )-, π 1 (Y log )- ∼ ∼ orbits of isomorphisms X Π → Π G X , Y Π → Π G Y discussed above — in X Π, Y Π of a verticial (respectively, cuspidal; nodal; edge-like) subgroup of Π G X , Π G Y as a verticial (respectively, cuspidal; nodal; edge- out out like) subgroup of X Π  X H, Y Π  Y H. We shall refer to a closed out out subgroup of X Π  X I, Y Π  Y I obtained by forming the normalizer out out out out in X Π  X I, Y Π  Y I [i.e., as opposed to X Π  X H, Y Π  Y H] of out a verticial (respectively, cuspidal; nodal; edge-like) subgroup of X Π  X out H, Y Π  Y H as a verticial (respectively, cuspidal; nodal; edge- out out like) I-decomposition subgroup of X Π  X H, Y Π  Y H. [In particular, for each  ∈ {X, Y }, it follows from [CmbGC], Proposition 1.2, (ii), that if  H satisfies condition (b) — which thus implies that  out Π =  Π   I — then it holds that a closed subgroup of  Π = 150  YUICHIRO HOSHI AND SHINICHI MOCHIZUKI out Π   I is a verticial (respectively, cuspidal; nodal; edge-like) out subgroup of  Π   H if and only if it is a verticial (respectively, out cuspidal; nodal; edge-like) I-decomposition subgroup of  Π   H.] Let out out ∼ α : X Π  X H −→ Y Π  Y H be an isomorphism of profinite groups. Then the following hold: (i) It holds that X H satisfies condition (a) (respectively, (b)) if and only if Y H satisfies condition (a) (respectively, (b)). (ii) The equality l X = l Y holds. (iii) The isomorphism α induces a bijection between the set of out verticial I-decomposition subgroups of X Π  X H and out the set of verticial I-decomposition subgroups of Y Π  Y H. (iv) The isomorphism α induces a bijection between the set of cuspidal (respectively, nodal; edge-like) I-decomposition out subgroups of X Π  X H and the set of cuspidal (respectively, out nodal; edge-like) I-decomposition subgroups of Y Π  Y H. (v) The isomorphism α restricts to an isomorphism out ( X Π  X H ⊇) X out ∼ out Π  X I −→ Y Π  Y I out (⊆ Y Π  Y H). (vi) There exists a positive integer n χ such that the diagram X out Y out ⊗nχ X Π  H −−−→ ⏐ ⏐ α   X χ G X H −−−→ Aut(G X ) −−− → Z ∗ l X Π  Y H −−−→ Y H −−−→ Aut(G Y ) −− ⊗n −→ Z ∗ l Y χ G χ Y — where χ G X , χ G Y are as in [CbTpI], Definition 3.8, (ii), and the right-hand vertical equality is the equality that arises from the equality l X = l Y of (ii) — commutes. (vii) The composite of the upper (respectively, lower) three horizon- tal arrows of the diagram of (vi) coincides with the composite of the upper (respectively, lower) three horizontal arrows of the COMBINATORIAL ANABELIAN TOPICS II 151 diagram X out Y out ⊗nχ X X Π  H −−−→ ⏐ ⏐ α   χ k X H −−−→ G k X −−− → Z ∗ l X Π  Y H −−−→ Y H −−−→ G k Y −− ⊗n −→ Z ∗ l Y χ k χ Y — where the integer n χ is the positive integer of (vi); the right- hand vertical equality is the equality that arises from the equal- ity l X = l Y of (ii); the third upper (respectively, lower) horizon- tal arrow is the n χ -th power of the l X - (respectively, l Y -) adic cyclotomic character χ k X of G k X (respectively, χ k Y of G k Y ). In particular, the diagram of the preceding display commutes. (viii) Suppose that one of the following three conditions is satisfied: (viii-1) Either X H or Y H satisfies condition (b). (viii-2) It holds that 0 ∈ {r X , r Y }. (viii-3) The isomorphism α induces a bijection between the set out of cuspidal subgroups of X Π  X H and the set of cus- out pidal subgroups of Y Π  Y H. Then the isomorphism α restricts to an isomorphism out ( X Π  X H ⊇) X ∼ Π −→ Y Π out (⊆ Y Π  Y H).  Proof. First, we verify assertions (i), (ii). Let  ∈ {X, Y }. Write Z (p ) for the pro-prime-to-p completion of the ring Z of rational integers. The following Facts are well-known: (1) The profinite group G k  is isomorphic to Z as an abstract profinite group. (2) The kernel of the natural surjection G log k   G k  admits a nat- (p  ) ural structure of free Z -module of rank 1. (3) The natural action by conjugation of G k  on the kernel of the natural surjection G log k   G k  is given by the cyclotomic char- acter [cf. (2)]. In particular, for each prime number q  = p, every maximal pro-q subgroup of G log k  admits a natural struc- ture of extension of Z q by Z q [cf. (1), (2)]. Moreover, the image of the action Z q → Aut(Z q ) = Z ∗ q determined by such an extension is open. Moreover, let us recall [cf., e.g., [AbsTpI], Proposition 2.3, (i)] that the following holds: 152 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (4) The pro-l  group  Π is nontrivial, center-free, and elastic [cf. [AbsTpI], Definition 1.1, (ii)]. Thus, we conclude that  H satisfies condition (b) if and only if the set of prime numbers q such that every maximal pro-q subgroup of  out Π   H is nonabelian is of cardinality 1. Moreover, the prime number l  may be characterized as the unique prime number q such that there out exists a maximal pro-q subgroup of  Π   H that is not isomorphic to a closed subgroup of an extension of Z q by Z q . This completes the proofs of assertions (i), (ii). In the remainder of the proof of Corollary 4.18, we shall write def l = l X = l Y [cf. assertion (ii)]. out Next, we verify assertion (iii). For  ∈ {X, Y } and J ⊆  Π   H an open subgroup, write J RTF for the maximal pro-RTF-quotient of the profinite group J [cf. [AbsTpI], def Proposition 1.2, (iv)];  Π J = J ∩  Π ⊆  Π;  H J ⊆  H for the image out of the composite J →  Π   H   H [so we have a commutative diagram of profinite groups 1 −−−→  Π J −−−→ ⏐ ⏐  J ⏐ ⏐   out 1 −−−→ −−−→  H J −−−→ 1 ⏐ ⏐  Π −−−→  Π   H −−−→  H −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are the natural inclusions]; G Jk  ⊆ G k  for the image of the composite out J →  Π   H   H → G log k   G k  ; (  Π J ) comb for the “combinatorial quotient” of  Π J , i.e., the quotient of  Π J by the normal closed subgroup normally topologically generated by the closed subgroups of  Π J obtained by forming the intersections of  Π J out with the verticial subgroups of  Π   H. Now we claim that the following assertion holds: out Claim 4.18.A: For  ∈ {X, Y } and J ⊆  Π   H an open subgroup, the quotient of J RTF by the image of the normal closed subgroup  Π J ⊆ J in J RTF is G Jk  . Indeed, this assertion follows immediately from Facts (1), (2), (3). Next, we claim that the following assertion holds: COMBINATORIAL ANABELIAN TOPICS II 153 out Claim 4.18.B: Let  ∈ {X, Y }, J ⊆  Π   H an open subgroup, Q a torsion-free abelian profinite group, and J → Q a homomorphism of profinite groups. Then the composite  Π J → J → Q factors through the natural surjection  Π J  (  Π J ) comb . To this end, let us first observe that since [it is well-known that] the image, in Z ∗ l , of the l-adic cyclotomic character of G k  is open, one verifies immediately that the image by the composite  Π J → J → out Q of any edge-like subgroup of  Π   H [i.e., any intersection of out  Π J with any edge-like subgroup of  Π   H] is trivial [cf., e.g., [CmbGC], Remark 1.1.3]. In a similar vein, it follows immediately from the “Riemann hypothesis for abelian varieties over finite fields” [cf., e.g., [Mumf], pp. 190-191] that the image by the composite  Π J → out J  Q of any verticial subgroup of  Π   H [i.e., any intersection of out  Π J with any verticial subgroup of  Π   H] is trivial. This completes the proof of Claim 4.18.B. Next, we claim that the following assertion holds: out Claim 4.18.C: For  ∈ {X, Y } and J ⊆  Π   H an open subgroup, the natural exact sequence 1 →  J Π → J →  H J → 1 fits into a commutative diagram of profinite groups 1 −−−→  Π J ⏐ ⏐  −−−→ J ⏐ ⏐  −−−→  H J −−−→ 1 ⏐ ⏐  (  Π J ) comb −−−→ J RTF −−−→ G Jk  −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are the natural surjections. Indeed, Claim 4.18.C follows immediately, in light of Claim 4.18.A, by applying Claim 4.18.B to the various RTF-subgroups of J [cf. [AbsTpI], Definition 1.1, (i)]. Next, we claim that the following assertion holds: Claim 4.18.D: For  ∈ {X, Y }, there exists an open out subgroup J 0 ⊆  Π   H that satisfies the following condition: For J ⊆ J 0 an arbitrary open subgroup, there exists an open subgroup J H † ⊆  H J such that if def we write J † = J ×  H J J H † , then the corresponding left- † hand lower horizontal arrow (  Π J ) comb → (J † ) RTF of the diagram of Claim 4.18.C is injective. 154 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI out To this end, let J 0 ⊆  Π   H be an open subgroup such that, for every open subgroup J ⊆ J 0 , the quotient (  Π J ) comb is a center-free free pro-l group [where we note that one verifies easily [cf. [CmbGC], Remark 1.1.3] that such a J 0 always exists]. Next, let us observe that, to verify Claim 4.18.D, we may assume without loss of gener- ality, by replacing  H J by a suitable open subgroup of  H J , that the outer action of J on (  Π J ) comb by conjugation is trivial [where we note that one verifies easily that such an open subgroup of  H J always ex- ists]. Since, as discussed above, (  Π J ) comb is center-free, if one writes def J comb = J/Ker(  Π J  (  Π J ) comb ), then this triviality implies that the inclusions (  Π J ) comb → J comb ← Z J comb ((  Π J ) comb ) [cf. the discussion entitled “Topological groups” in [CbTpI], §0] deter- mine an isomorphism ∼ (  Π J ) comb × Z J comb ((  Π J ) comb ) −→ J comb . On the other hand, since (  Π J ) comb is a free pro-l group, the nat- ural surjection (  Π J ) comb  ((  Π J ) comb ) RTF is an isomorphism. In ∼ particular, the composite of natural homomorphisms (  Π J ) comb → ((  Π J ) comb ) RTF → (J comb ) RTF is injective. Thus, since the natural surjection J  (J comb ) RTF factors through J RTF , Claim 4.18.D follows immediately. This completes the proof of Claim 4.18.D. Next, we claim that the following assertion holds: out Claim 4.18.E: Let  ∈ {X, Y } and A ⊆  Π   H a closed subgroup. Then the following two conditions are equivalent: out (E-1) The closed subgroup A ⊆  Π   H is contained out in a verticial I-decomposition subgroup of  Π   H. out (E-2) For J ⊆  Π   H an arbitrary open subgroup, the composite A ∩ J → J  J RTF is trivial. To this end, let us first observe that the implication (E-1) ⇒ (E-2) follows immediately from Claim 4.18.C, together with Facts (1), (2), (3). On the other hand, by applying Claims 4.18.C, 4.18.D to the out various open subgroups of  Π   H for each  ∈ {X, Y }, one verifies immediately from Proposition 1.5 that the implication (E-2) ⇒ (E- 1) holds. This completes the proof of Claim 4.18.E. On the other hand, since any inclusion of verticial I-decomposition subgroups is an equality [cf. [CmbGC], Proposition 1.2, (i), (ii)], assertion (iii) follows immediately from Claim 4.18.E. This completes the proof of assertion (iii). COMBINATORIAL ANABELIAN TOPICS II 155 Next, we verify assertion (iv). We begin the proof of assertion (iv) with the following claim: Claim 4.18.F: Let  ∈ {X, Y }. Suppose that  log is a smooth log curve over (Spec k  ) log [cf. the discussion entitled “Curves” in [CbTpI], §0]. Then the inclusions  out out Π →  Π   I ← Z(  Π   I) [cf. the discussion entitled “Topological groups” in [CbTpI], §0] determine an isomorphism  out ∼ out Π × Z(  Π   I) −→  Π   I. out out Moreover, the composite Z(  Π   I) →  Π   I   I is an isomorphism. In particular, if  H satisfies out condition (a) (respectively, (b)), then Z(  Π   I)  admits a structure of free Z (p ) -module of rank 1 (re- spectively, is trivial) [cf. Fact (2)]. Indeed, since [we have assumed that]  log is a smooth log curve over (Spec k  ) log , this assertion follows immediately from the slimness of  Π [cf. [CmbGC], Remark 1.1.3], together with the various definitions involved. Next, let us observe that it follows from [CmbGC], Proposition 1.2, (ii), that, (5) for each  ∈ {X, Y }, if A is a VCN-subgroup of  Π, then out the intersection of  Π with the normalizer, in  Π   I, of A coincides with A. Moreover, let us also observe that it follows from [NodNon], Remark 2.4.2; [NodNon], Remark 2.7.1, that, (6) for each  ∈ {X, Y }, any inclusion of VCN-subgroups of  Π out gives rise to an inclusion of the normalizers, in  Π   I, of the respective VCN-subgroups. Next, we claim that the following assertion holds: Claim 4.18.G: The isomorphism α induces a bijection between the set of edge-like I-decomposition subgroups out of X Π  X H and the set of edge-like I-decomposition out subgroups of Y Π  Y H. To this end, let us first observe that it follows immediately — in light of Facts (5), (6) — from assertion (iii) that, to verify Claim 4.18.G, we out may assume without loss of generality — by replacing  Π   H by out the normalizer, in  Π   H, of a verticial I-decomposition subgroup 156 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI out of  Π   H for each  ∈ {X, Y } — that X log , Y log are smooth log curves over (Spec k X ) log , (Spec k Y ) log , and that the isomorphism α out out out restricts to an isomorphism of X Π  X I (⊆ X Π  X H) with Y Π  Y I out (⊆ Y Π  Y H). Next, let us observe that if X H, hence also Y H [cf. assertion (i)], satisfies condition (b), then since [it is well-known that] the image, in Z ∗ l , of the l-adic cyclotomic character of G k  is open for each  ∈ {X, Y }, Claim 4.18.G follows immediately from [CmbGC], Corollary 2.7, (i). Thus, in the remainder of the proof of Claim 4.18.G, we may assume without loss of generality that X H, hence also Y H [cf. assertion (i)], satisfies condition (a). Then, by applying a similar argument to the argument in the proof of Claim 4.18.G in the case where X H satisfies condition (b) to the isomorphism out out ∼ out out ( X Π  X H)/Z( X Π  X I) −→ ( Y Π  Y H)/Z( Y Π  Y I) induced by α [cf. Claim 4.18.F], we conclude that this induced iso- morphism determines a bijection between the set of images of edge-like out out out subgroups of X Π  X H in the quotient ( X Π  X H)/Z( X Π  X I) and out the set of images of edge-like subgroups of Y Π  Y H in the quotient out out ( Y Π  Y H)/Z( Y Π  Y I). Now let us observe that it follows imme- diately from Claim 4.18.F and [CmbGC], Proposition 1.2, (ii), that, out for each  ∈ {X, Y } and each edge-like subgroup A ⊆  Π   H, the out edge-like I-decomposition subgroup of  Π   H obtained by forming out the normalizer of A in  Π   I coincides with the inverse image by out out out the natural surjection  Π   H  (  Π   H)/Z(  Π   I) of the out out image of A in (  Π   H)/Z(  Π   I). Thus, Claim 4.18.G holds. This completes the proof of Claim 4.18.G. On the other hand, assertion (iv) follows — in light of Facts (5), (6) — from assertion (iii), Claim 4.18.G, and [CmbGC], Proposition 1.5, (i). This completes the proof of assertion (iv). Next, we verify assertions (v), (vi), (vii). First, we observe that assertion (vii) is a formal consequence of assertion (vi), together with [AbsCsp], Proposition 1.2, (ii); [CbTpI], Corollary 3.9, (ii), (iii). Now out suppose that there is no nodal subgroup of X Π  X H, hence also [cf. out assertion (iv)] of Y Π  Y H. Then assertion (v) follows from assertion (iii). Moreover, by considering, for each  ∈ {X, Y }, the cyclotome obtained by applying the construction of “Λ v ” of [CbTpI], Definition 3.8, (i), to the collection of data consisting of COMBINATORIAL ANABELIAN TOPICS II out 157 out • the profinite group (  Π   I)/Z(  Π   I) and out out • the various images in (  Π   I)/Z(  Π   I) of the edge-like out I-decomposition subgroups of  Π   H, one verifies immediately from assertions (iv), (v), together with Claim 4.18.F, that assertion (vi) [i.e., in the case where one takes “n χ ” in the statement of assertion (vi) to be 1], hence also assertion (vii), holds. Thus, in the remainder of the proofs of assertions (v), (vi), (vii), out we may assume without loss of generality that both X Π  X H and Y out Π  Y H have a nodal subgroup. Then one verifies immediately from assertions (iii), (iv) [cf. also Facts (5), (6)], together with Lemma 4.19 below [cf. [NodNon], Definition 2.4, (i); [NodNon], Remark 2.4.2], that assertion (vi), hence also assertion (vii), holds. On the other hand, for each  ∈ {X, Y }, we conclude from Fact (2) in the proof of Corollary 4.16 that (7) if we write out out (  Π   I) (l) ⊆  Π   I for the [unique — cf. Fact (2)] maximal pro-l subgroup of  out out out Π   I, then the closed subgroup (  Π   I) (l) ⊆ (  Π   out out I ⊆)  Π   H coincides with the closed subgroup of  Π   H obtained by forming the unique maximal pro-l subgroup of the kernel of the composite of the relevant [i.e., upper if  = X; lower if  = Y ] three horizontal arrows of the diagram of assertion (vii). Moreover, we also conclude immediately from Facts (1), (2), (3) that, for each  ∈ {X, Y }, (8) the kernel of the composite  out out out out out Π   H   Π   H/(  Π   I) (l)  (  Π   H/(  Π   I) (l) ) RTF out coincides with the closed subgroup  Π   I. In particular, it follows from assertion (vii) and Facts (7), (8) that the out out isomorphism α restricts to an isomorphism of X Π  X I (⊆ X Π  X H) out out with Y Π  Y I (⊆ Y Π  Y H), as desired. This completes the proof of assertion (v). Finally, we verify assertion (viii). If condition (viii-1) is satisfied, out then since [it follows from assertion (i) that]  Π =  Π   I for each  ∈ {X, Y }, assertion (viii) follows from assertion (v). Thus, in the 158 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI remainder of the proof of assertion (viii), we suppose that both X H and Y H satisfy condition (a). Next, suppose that condition (viii-2) is satisfied. Then it follows from assertion (iv) that (r X , r Y ) = (0, 0). Write  (l) I ⊆  I for the [unique — cf. Fact (2)] maximal pro-l subgroup of  I. Then one verifies easily that one may naturally regard  I (l) as a quotient of out (  Π   I) (l) [cf. Fact (7)], and, moreover, that the closed subgroup  out Π of  Π   H coincides with the kernel of the natural surjection out (  Π   I) (l)   I (l) . In particular, it follows from assertions (v), (vii) that, to verify assertion (viii) in the case where condition (viii-2) is satisfied, it suffices to verify the following assertion: out Claim 4.18.H: Let  ∈ {X, Y } and (  Π   I) (l)  A out a quotient of (  Π   I) (l) . Then it holds that this out quotient (  Π   I) (l)  A coincides with the quo- out tient (  Π   I) (l)   I (l) if and only if the following three conditions are satisfied: (H-1) The profinite group A is isomorphic to Z l as an abstract profinite group. out (H-2) The kernel of the surjection (  Π   I) (l)  A out is normal in  Π   H. Thus, the outer action out out of  Π   H on (  Π   I) (l) by conjugation out induces an action [cf. (H-1)] of  Π   H on the quotient A. Moreover, the resulting character out ρ A :  Π   H → Aut(A) = Z ∗ l [cf. (H-1)] is out out trivial on  Π   I ⊆  Π   H. (H-3) The n χ -th power of the character ρ A of (H-2) co- incides with the composite of the relevant [i.e., upper if  = X; lower if  = Y ] three horizontal arrows of the diagram of assertion (vii). First, let us observe that it follows from Facts (2), (3) that the quo- out tient (  Π   I) (l)   I (l) satisfies the three conditions in the state- ment of Claim 4.18.H. Next, let us observe that it follows immedi- ately from [CmbGC], Propositions 1.3, 2.6, that if a given quotient out (  Π   I) (l)  A satisfies conditions (H-1), (H-2), then the image in A of an arbitrary nodal [or, equivalently, edge-like — cf. the equality COMBINATORIAL ANABELIAN TOPICS II 159 out (r X , r Y ) = (0, 0) discussed above] subgroup of  Π   H is trivial. Thus, it follows immediately from the “Riemann hypothesis for abelian varieties over finite fields” [cf., e.g., [Mumf], pp. 190-191], together with Fact (3) and condition (H-3), that Claim 4.18.H holds. This com- pletes the proof of Claim 4.18.H, hence also of assertion (viii) in the case where condition (viii-2) is satisfied. Finally, one may verify assertion (viii) in the case where condition (viii-3) is satisfied by applying assertion (viii) in the case where condi- tion (viii-2) is satisfied. Indeed, let us first observe that it follows im- mediately from Fact (1) [which implies that G k X , G k Y are torsion-free] out that we may assume without loss of generality, by replacing  Π   H out by a suitable open subgroup of  Π   H for each  ∈ {X, Y }, that g X , g Y ≥ 2. Then we may assume without loss of generality, by re- out out placing  Π   H by the quotient of  Π   H by the normal closed subgroup normally topologically generated by the cuspidal subgroups for each  ∈ {X, Y }, that (r X , r Y ) = (0, 0) [cf. (viii-3)]. Thus, it fol- lows from assertion (viii) in the case where condition (viii-2) is satisfied that assertion (viii) holds. This completes the proof of assertion (viii), hence also of Corollary 4.18.  Remark 4.18.1. In the situation of Corollary 4.18, (viii), if one omits the assumption that one of the conditions (viii-1), (viii-2), and (viii-3) holds, then the conclusion of Corollary 4.18, (viii), no longer holds in general. Indeed: (i) First, we consider the case of a smooth log curve [cf. the dis- cussion entitled “Curves” in [CbTpI], §0]. In the situation def of Corollary 4.18, write l = l X . Let T log be a tripod over (Spec k X ) log [cf. the discussion entitled “Curves” in [CbTpI], §0] such that the natural action of G log k X on the set of cusps of T log is trivial. Then, by taking “ T H” to be G log k X , we obtain a out profinite group T Π  T H. In the remainder of the discussion of the present (i), out we construct an automorphism of T Π  T H that out does not preserve the closed subgroup T Π ⊆ T Π  T H. out out Let C ⊆ T Π  T H be a cuspidal subgroup of T Π  T H. def out out Write Z = Z( T Π  T I) for the center of T Π  T I and out I C ⊆ T Π  T H for the cuspidal I-decomposition subgroup of 160 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI T out out Π  T H obtained by forming the normalizer in T Π  T I of C. Then out out (i-a) the natural inclusions T Π → T Π  T I and Z → T Π  T I out ∼ determine an isomorphism T Π× Z → T Π  T I [cf. Claim 4.18.F]. Moreover, the natural inclusions C → I C and ∼ Z → I C determine an isomorphism C ×Z → I C [cf. Claim 4.18.F; [CmbGC], Proposition 1.2, (ii)]. Moreover, it is well-known that the following assertions hold: (i-b) The unique maximal pro-l subgroup of Z admits a struc- ture of free Z l -module of rank 1 [cf. Claim 4.18.F]. More- over, the natural action of G k X on this unique maximal out ∼ pro-l subgroup of Z (= {0} × Z ⊆ T Π ab × Z → (  Π   ab I) ) [cf. (i-a)] induced by the natural outer action of out G k X on T Π  T I is given by the l-adic cyclotomic char- acter [cf. Fact (3) in the proof of Corollary 4.18]. (i-c) The pro-l group T Π ab admits a structure of free Z l -module of rank 2. Moreover, the natural action of G k X on T Π ab out ∼ (= T Π ab × {0} ⊆ T Π ab × Z → (  Π   I) ab ) [cf. (i-a)] out induced by the natural outer action of G k X on T Π  T I is given by the l-adic cyclotomic character. Thus, since C admits a structure of free Z l -module of rank 1 [cf. [CmbGC], Remark 1.1.3], there exists a nontrivial homo- morphism φ : T Π ( T Π ab ) → Z whose kernel is topologically normally generated by C. Now write α I for the automorphism out ∼ of the profinite group T Π × Z ( → T Π  T I) [cf. (i-a)] given by mapping T Π × Z  (σ, z) → (σ, z · φ(σ)) ∈ T Π × Z. Next, let out def us observe that the composite H C = N T out T (C) → T Π  Π  T H H  G k X is surjective, with kernel equal to I C . Thus, it follows from Fact (1) in the proof of Corollary 4.18 that this composite H C  G k X admits a section, which determines an isomorphism out ∼ out ( T Π  T I)  G k X −→ T Π  T H. Let us fix such a section. Next, observe that it follows from (i- b), (i-c) that the above automorphism α I is compatible with the out action of G k X on T Π  T I determined by the fixed section of H C  G k X . Thus, we conclude that the above automorphism out out α I of T Π  T I extends to an automorphism α of T Π  T H COMBINATORIAL ANABELIAN TOPICS II 161 that preserves and induces the identity automorphism on the image of the fixed section of H C  G k X . Now let us observe that it is immediate that out α I , hence also α, does not preserve T Π ⊆ T Π  T I, as desired. Let us also observe that since C ⊆ T Π is con- tained in the kernel of φ, it follows from (i-a) that α I pre- serves and induces the identity automorphism on the cuspidal out I-decomposition subgroup I C ⊆ T Π  T I. In particular, we conclude immediately that out (i-d) the automorphism α of T Π  T H preserves and induces the identity automorphism on H C . (ii) Next, we consider the case of a singular stable log curve [i.e., a stable log curve that is not smooth]. In the situation of (i), let W log be a stable log curve over (Spec k X ) log such that • W log has precisely two irreducible components each of which is a tripod, • W log has a single node, and, moreover, log • the natural action of G log k X on the dual semi-graph of W is trivial. [Thus, W log is of type (0, 4).] Then, by taking “ W H” to be G log k X , out we obtain a profinite group W Π  W H [cf. the situation and notational conventions of Corollary 4.18]. In the remainder of the discussion of the present (ii), out we construct an automorphism of W Π  W H that out does not preserve the closed subgroup W Π ⊆ W Π  W H. Write v 1 , v 2 for the distinct two irreducible components of out W log . Let V 1 , V 2 ⊆ W Π ⊆ W Π  W H be verticial subgroups of W out def Π  W H associated to v 1 , v 2 such that N = V 1 ∩ V 2  = {1}, which thus [cf. [NodNon], Lemma 1.9, (i)] implies that N is a out nodal subgroup of W Π  W H. For each i ∈ {1, 2}, write def H V i = N W out W (V i ), Π  H def H N = N W out W (N ). Π  H Then one verifies immediately [cf. [CmbGC], Proposition 1.2, (ii)] that 162 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (ii-a) there exists a commutative diagram of profinite groups N ⊆ ↓ C V i ⊆ ↓ ⊆ T Π ⊆ T H V i ⊇ H N ↓ ↓ out Π  T H ⊇ H C — where the horizontal arrows are the natural inclusions, and the vertical arrows are isomorphisms. Moreover, it follows immediately from a similar argument to the argument applied in the proof of [CmbCsp], Proposition 1.5, (iii) [i.e., in essence, from the evident analogue for semi- graphs of anabelioids of the “van Kampen Theorem” in ele- mentary algebraic topology], that (ii-b) the natural inclusions out H V 1 → W Π  W H ← H V 2 determine an isomorphism ∼ out lim (H V 1 ← H N → H V 2 ) −→ W Π  W H −→ — where the inductive limit is taken in the category of profinite groups — which restricts to an isomorphism of closed subgroups ∼ lim (V 1 ← N → V 2 ) −→ W Π −→ — where the inductive limit is taken in the category of profinite groups. On the other hand, it follows from (i-d) and (ii-a) that, for each i ∈ {1, 2}, α determines an automorphism β i of H V i that • does not preserve V i ⊆ H V i but • preserves and induces the identity automorphism on the closed subgroup H N ⊆ H V i . Thus, by (ii-b), β 1 and β 2 determine an automorphism γ of out W Π  W H that does not preserve the closed subgroup W Π ⊆ W Π  W H, as desired. out Lemma 4.19 (An explicit description of a power of the cyclo- tomic character). Let J be a profinite group, ρ J : J → Aut(G 0 ) a continuous homomorphism, and I ⊆ J a normal closed subgroup of J such that either COMBINATORIAL ANABELIAN TOPICS II 163 ρ J (a) the composite I → J → Aut(G 0 ) is of SNN-type [cf. [NodNon], Definition 2.4, (iii)], or (b) I = {1}. Write def out def out Π I = Π G 0  I ⊆ Π J = Π G 0  J [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. Thus, we have a commutative diagram of profinite groups 1 −−−→ Π G 0 −−−→ Π I −−−→ I −−−→ 1 ⏐ ⏐ ⏐ ⏐   1 −−−→ Π G 0 −−−→ Π J −−−→ J −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are the natural inclusions. Write G  0 → G 0 for the universal covering of G 0 corresponding to Π G 0 . Let e 0 be a node of G  0 . Write Π e 0 ⊆ Π G 0 for the nodal subgroup associated to e 0 . Write Π e 0 ,J ⊆ Π J for the [necessarily open] subgroup consisting of the elements σ ∈ Π J such that the natural action of σ on the underlying semi-graph G 0 of G 0 stabilizes the two branches of the node e 0 (G 0 ) of G 0 determined by e 0 . Then the following hold: (i) Let N be a positive integer and γ an element of Π J . Then there exists a collection of data as follows • a normal open subgroup H ⊆ Π J of Π J , • a positive integer m, • verticial subgroups Π v 0 , Π v 1 , . . . , Π v m−1 ⊆ Π G 0 of Π G 0 asso- ciated to vertices v 0 , v 1 , . . . , v m−1 of G  0 , respectively, and • nodal subgroups Π e 1 , . . . , Π e m ⊆ Π G 0 of Π G 0 associated to nodes e 1 , . . . , e m of G  0 , respectively, such that if we write def D e j = N Π I (Π e j ) for each j ∈ {0, 1, . . . , m} [cf. [NodNon], Definition 2.2, (iii)], then (1) the inclusions Π e 0 ⊆ Π v 0 , Π e m ⊆ Π v m−1 [which imply that e 0 , e m abut to v 0 , v m−1 , respectively — cf. [NodNon], Lemma 1.7] hold, (2) if m ≥ 2, then, for every j ∈ {1, . . . , m − 1}, the inclusion Π e j ⊆ Π v j−1 ∩ Π v j [which implies that e j abuts to v j−1 and v j — cf. [NodNon], Lemma 1.7] holds, (3) the quotient D e 0  D e 0 ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 ) 164 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI  ( Z Σ 0 /N Z Σ 0 ) × ( Z Σ 0 /N Z Σ 0 ) if (a) is satisfied Z Σ 0 /N Z Σ 0 if (b) is satisfied [cf. [CmbGC], Remark 1.1.3; [NodNon], Lemma 2.5, (i); [NodNon], Remark 2.7.1] of D e 0 factors through the quo- tient of D e 0 determined by the composite D e 0 → Π J  Π J /H, and, moreover, (4) the image of D e m ⊆ Π J in Π J /H coincides with the im- age of γ · D e 0 · γ −1 ⊆ Π J in Π J /H. For each j ∈ {0, 1, . . . , m − 1}, write ∼ = def def D v j = N Π I (Π v j ) ⊇ I v j = Z Π I (Π v j ) = Z(D v j ) [cf. [NodNon], Definition 2.2, (i); [NodNon], Lemma 2.5, (i); [NodNon], Remark 2.7.1; [CmbGC], Remark 1.1.3]; b j,j , b j+1,j for the respective branches of the nodes e j , e j+1 that abut to the vertex v j determined by the inclusions Π e j ⊆ Π v j , Π e j+1 ⊆ Π v j [cf. (1), (2)]. Thus, for j ∈ {0, 1, . . . , m} and s ∈ {0, 1, . . . , m − 1} such that s ∈ {j − 1, j}, it follows from [NodNon], Remark 2.7.1, that we have natural inclusions I v s ⊆ D e j ⊆ D v s ∪ ∪ Π e j ⊆ Π v s , which determine a commutative diagram of profinite groups D e j /I v s −−−→ D v s /I v s   ⏐ ⏐   ⏐ ⏐ Π e j −−−→ Π v s — where the horizontal arrows are the natural inclusions, and the vertical arrows are isomorphisms. (ii) In the situation of (i), by applying the construction of “Λ v ” of [CbTpI], Definition 3.8, (i), to the collection of data consisting of • the profinite group D v s /I v s and • the various images in D v s /I v s , by the right-hand vertical ∼ isomorphism Π v s → D v s /I v s of the final display of (i), of the edge-like subgroups of Π G 0 contained in Π v s , one may construct a cyclotome Λ(D v s /I v s ). Moreover, by applying the construction of “syn b ” of [CbTpI], Corollary 3.9, (v), to the collection of data consisting of • the profinite groups D e j /I v s , D v s /I v s , COMBINATORIAL ANABELIAN TOPICS II 165 • the various images in D v s /I v s , by the right-hand vertical ∼ isomorphism Π v s → D v s /I v s of the final display of (i), of the edge-like subgroups of Π G 0 contained in Π v s , and • the upper horizontal arrow D e j /I v s → D v s /I v s of the final display of (i) [i.e., that corresponds the branch b j,s ], one may construct an isomorphism ∼ syn b j,s : D e j /I v s −→ Λ(D v s /I v s ). Write  def M v s = M e j I v s ⊗ Z  Σ0 Λ(D v s /I v s ) if (a) is satisfied if (b) is satisfied Λ(D v s /I v s )  det(D e j ) if (a) is satisfied def = D e j /I v s if (b) is satisfied — where the “det” is taken with respect to the structure of free Z Σ 0 -module of finite rank of the profinite group D e j ; we observe that if condition (a) is satisfied, then the exact sequence of free Z Σ 0 -modules of finite rank 1 −→ I v s −→ D e j −→ D e j /I v s −→ 1 yields a natural identification M e j = I v s ⊗ Z  Σ0 (D e j /I v s ) of Z Σ 0 -modules [cf. [CmbGC], Remark 1.1.3; [NodNon], Lemma 2.5, (i); [NodNon], Remark 2.7.1]; we observe that if condition (b) is satisfied, then since I v s = {1}, we have a natural isomor- ∼ phism D e j → D e j /I v s = M e j . If condition (a) is satisfied, then let us write M ∼ syn b j,s : M e j = I v s ⊗ Z  Σ0 (D e j /I v s ) −→ I v s ⊗ Z  Σ0 Λ(D v s /I v s ) = M v s for the isomorphism determined by the above isomorphism syn b j,s . If condition (b) is satisfied, then let us write M ∼ def syn b j,s = syn b j,s : M e j = D e j /I v s −→ Λ(D v s /I v s ) = M v s . def (iii) In the situation of (ii), write n 0 = 2 (respectively, 1) if condi- tion (a) (respectively, (b)) is satisfied. Write Φ N (γ) ∈ Aut(M e 0 ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 )) = ( Z Σ 0 /N Z Σ 0 ) ∗ for the automorphism of the free Z Σ 0 -module [of rank one] M e 0 ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 ) obtained by forming the composite of the isomorphism ∼ M e 0 ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 ) −→ M e m ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 ) 166 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI determined by conjugating by γ ∈ Π J [cf. conditions (3), (4) in (i)] with the isomorphism ∼ M e m ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 ) −→ M e 0 ⊗ Z  Σ0 ( Z Σ 0 /N Z Σ 0 ) determined by the inverse of the composite M syn −1 b 1,0 M syn b 0,0 ∼ ∼ M syn −1 b 2,1 M syn b 1,1 ∼ ∼ M e 0 −→ M v 0 −→ M e 1 −→ M v 1 −→ M syn −1 bm,m−1 M syn bm−1,m−1 ∼ ∼ ... −→ M v m−1 −→ M e m . Suppose that γ ∈ Π e 0 ,J . Then the image of γ by the composite ⊗2n 0 χ G ρ J 0 Π J −→ J −→ Aut(G 0 ) −→ ( Z Σ 0 ) ∗ −→ ( Z Σ 0 /N Z Σ 0 ) ∗ [cf. [CbTpI], Definition 3.8, (ii)] coincides with Φ N (γ) 2 ∈ ( Z Σ 0 /N Z Σ 0 ) ∗ . (iv) Let ρ : Π J → ( Z Σ 0 ) ∗ be a character [i.e., a continuous homo- morphism] and n ρ a positive integer divisible by 2[Π J : Π e 0 ,J ]. Suppose that, for each positive integer N  and each γ  ∈ Π e 0 ,J , the image of ρ(γ  ) 2 ∈ ( Z Σ 0 ) ∗ in ( Z Σ 0 /N  Z Σ 0 ) ∗ coincides with Φ N  (γ  ) 2 ∈ ( Z Σ 0 /N  Z Σ 0 ) ∗ [cf. (iii)]. Then the n ρ -th power of the character ρ coincides with the n ρ -th power of the character obtained by forming the composite ρ ⊗n 0 χ G J 0 Π J −→ J −→ Aut(G 0 ) −→ ( Z Σ 0 ) ∗ [cf. (iii)]. Proof. Assertions (i), (ii) follow immediately from the various defini- tions involved. Next, we verify assertion (iii). Let us first observe that it follows immediately from the various definitions involved that there exist δ ∈ Π G 0 ⊆ Π e 0 ,J and ∈ N Π J (Π e 0 ) ∩ Π e 0 ,J (⊆ N Π J (D e 0 ) ∩ Π e 0 ,J ) such that γ = δ · . Now one verifies immediately from [CbTpI], Corol- lary 3.9, (ii), (v); [CbTpI], Corollary 5.9, (ii), that the action of on M e 0 by conjugation is given by multiplication by χ G 0 ( ) n 0 . Moreover, let us observe that one verifies easily that the collection of data of as- sertion (i) [i.e., associated to γ] satisfies conditions (1), (2), (3), (4) in assertion (i) in the case where we take “γ” to be δ. Also, let us observe that the image of δ by the composite ρ J ⊗n 0 χ G 0 Π J −→ J −→ Aut(G 0 ) −→ ( Z Σ 0 ) ∗ −→ ( Z Σ 0 /N Z Σ 0 ) ∗ is trivial. Thus, assertion (iii) follows immediately from [CbTpI], Corol- lary 3.9, (ii), (v), (vi). This completes the proof of assertion (iii). As- sertion (iv) is a formal consequence of assertion (iii). This completes the proof of Lemma 4.19.  COMBINATORIAL ANABELIAN TOPICS II 167 References [Asd] M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), 123-147. [HT] H. Hamidi-Tehrani, Groups generated by positive multi-twists and the fake lantern problem, Algebr. Geom. Topol. 2 (2002), 1155-1178. [Hsh] Y. Hoshi, Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic curves over finite fields, Publ. Res. Inst. Math. Sci. 45 (2009), 661-744. [NodNon] Y. Hoshi and S. Mochizuki, On the combinatorial anabelian geometry of nodally nondegenerate outer representations, Hiroshima Math. J. 41 (2011), 275-342. [CbTpI] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial an- abelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn twists, Galois-Teichmüller Theory and Arithmetic Geometry, 659-811, Adv. Stud. Pure Math., 63, Math. Soc. Japan, Tokyo, 2012. [Ishi] A. Ishida, The structure of subgroup of mapping class groups generated by two Dehn twists, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 240-241. [Lch] P. Lochak, Results and conjectures in profinite Teichmüller theory, Galois- Teichmüller Theory and Arithmetic Geometry, 263-335, Adv. Stud. Pure Math., 63, Math. Soc. Japan, Tokyo, 2012. [LocAn] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), 319-423. [ExtFam] S. Mochizuki, Extending families of curves over log regular schemes, J. Reine Angew. Math. 511 (1999), 43-71. [SemiAn] S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), 221-322. [CmbGC] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math. J. 59 (2007), 455-479. [AbsCsp] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), 451-539. [AbsTpI] S. Mochizuki, Topics in Absolute Anabelian Geometry I: Generalities, J. Math. Sci. Univ. Tokyo. 19 (2012), 139-242. [CmbCsp] S. Mochizuki, On the Combinatorial Cuspidalization of Hyperbolic Curves, Osaka J. Math. 47 (2010), 651-715. [MzTa] S. Mochizuki and A. Tamagawa, The Algebraic and Anabelian Geometry of Configuration Spaces, Hokkaido Math. J. 37 (2008), 75-131. [Mumf] D. Mumford, Abelian varieties, with appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the second (1974) edition. Tata Institute of Fundamental Research Studies in Mathematics, 5. Published for the Tata In- stitute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. [RZ] L. Ribes and P. Zalesskii, Profinite groups, Second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 40. Springer-Verlag, Berlin, 2010. [Wkb] Y. Wakabayashi, On the cuspidalization problem for hyperbolic curves over finite fields, Kyoto J. Math. 56 (2016), 125-164. [SGA1] Revêtements étales et groupe fondamental (SGA 1), Séminaire de Géométrie Algébrique du Bois Marie 1960-1961, directed by A. Grothendieck, with two papers by M. Raynaud, Documents Mathématiques (Paris), 3, Société Mathématique de France, Paris, 2003.